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#### Question 34:

Prove each of the following identities:

#### Question 1:

(i) (1 − cos2θ) cosec2θ = 1
(ii) (1 + cot2θ) sin2θ = 1

#### Question 2:

(i) (sec2θ − 1) cot2θ = 1
(ii) (sec2θ − 1) (cosec2θ − 1) = 1
(iii) (1− cos2θ) sec2θ = tan2θ

#### Question 3:

(i) ${\mathrm{sin}}^{2}\mathrm{\theta }+\frac{1}{\left(1+{\mathrm{tan}}^{2}\mathrm{\theta }\right)}=1$
(ii) $\frac{1}{\left(1+{\mathrm{tan}}^{2}\mathrm{\theta }\right)}+\frac{1}{\left(1+{\mathrm{cot}}^{2}\mathrm{\theta }\right)}=1$

#### Question 4:

(i) (1 + cos θ) (1 − cos θ) (1 + cot2θ) = 1
(ii) cosec θ (1 + cos θ) (cosec θ − cot θ) = 1

#### Question 5:

(i) $1+\frac{{\mathrm{cot}}^{2}\mathrm{\theta }}{\left(1+\mathrm{cosec\theta }\right)}=\mathrm{cosec\theta }$
(ii) $1+\frac{{\mathrm{tan}}^{2}\mathrm{\theta }}{\left(1+\mathrm{sec\theta }\right)}=\mathrm{sec\theta }$

#### Question 6:

(i) sec θ (1 − sin θ) (sec θ + tan θ) = 1
(ii) sin θ(1 + tan θ) + cos θ(1 + cot θ) = (sec θ + cosec θ)

#### Question 7:

$\frac{\mathrm{sin\theta }}{1+\mathrm{cos\theta }}+\frac{\left(1+\mathrm{cos\theta }\right)}{\mathrm{sin\theta }}=2\mathrm{cosec\theta }$

Hence, LHS = RHS

#### Question 9:

Hence, L.H.S. = R.H.S.

#### Question 10:

$\frac{\mathrm{cos\theta }}{\left(1-\mathrm{tan\theta }\right)}+\frac{{\mathrm{sin}}^{2}\mathrm{\theta }}{\left(\mathrm{cos\theta }-\mathrm{sin\theta }\right)}=\left(\mathrm{cos\theta }+\mathrm{sin\theta }\right)$

Hence, LHS = RHS

#### Question 11:

$\frac{{\mathrm{tan}}^{2}\mathrm{\theta }}{\left(1+{\mathrm{tan}}^{2}\mathrm{\theta }\right)}+\frac{{\mathrm{cot}}^{2}\mathrm{\theta }}{\left(1+{\mathrm{cot}}^{2}\mathrm{\theta }\right)}=1$

Hence, LHS = RHS

#### Question 12:

$\frac{\left(1+{\mathrm{tan}}^{2}\mathrm{\theta }\right)\mathrm{cot\theta }}{{\mathrm{cosec}}^{2}\mathrm{\theta }}=\mathrm{tan}\theta$

Hence, L.H.S. = R.H.S.

#### Question 13:

$\left(1+{\mathrm{tan}}^{2}\mathrm{\theta }\right)\left(1+{\mathrm{cot}}^{2}\mathrm{\theta }\right)=\frac{1}{\left({\mathrm{sin}}^{2}\mathrm{\theta }-{\mathrm{sin}}^{4}\mathrm{\theta }\right)}$

Hence, LHS = RHS

Hence, LHS = RHS

#### Question 15:

(i) ${\mathrm{sin}}^{6}\mathrm{\theta }+{\mathrm{cos}}^{6}\mathrm{\theta }=1-3{\mathrm{sin}}^{2}{\mathrm{\theta cos}}^{2}\mathrm{\theta }$
(ii) ${\mathrm{sin}}^{2}\mathrm{\theta }+{\mathrm{cos}}^{4}\mathrm{\theta }={\mathrm{cos}}^{2}\mathrm{\theta }+{\mathrm{sin}}^{4}\mathrm{\theta }$
(iii) ${\mathrm{cosec}}^{4}\mathrm{\theta }-{\mathrm{cosec}}^{2}\mathrm{\theta }={\mathrm{cot}}^{4}\mathrm{\theta }+{\mathrm{cot}}^{2}\mathrm{\theta }$

#### Question 16:

(i) $\frac{1-\mathrm{sin\theta }}{1+\mathrm{sin\theta }}=\left(\mathrm{sec\theta }-\mathrm{tan\theta }{\right)}^{2}$
(ii) $\frac{1+\mathrm{cos\theta }}{1-\mathrm{cos\theta }}=\left(\mathrm{cosec\theta }+\mathrm{cot\theta }{\right)}^{2}$

#### Question 17:

$\frac{1+{\mathrm{tan}}^{2}\mathrm{\theta }}{1+{\mathrm{cot}}^{2}\mathrm{\theta }}={\left(\frac{1+\mathrm{tan\theta }}{1-\mathrm{cot\theta }}\right)}^{2}={\mathrm{tan}}^{2}\mathrm{\theta }$

∴ LHS = RHS
Hence proved.

#### Question 18:

(i) $\frac{1-{\mathrm{tan}}^{2}\mathrm{\theta }}{1+{\mathrm{tan}}^{2}\mathrm{\theta }}=\left({\mathrm{cos}}^{2}\mathrm{\theta }-{\mathrm{sin}}^{2}\mathrm{\theta }\right)$
(ii) $\frac{1-{\mathrm{tan}}^{2}\mathrm{\theta }}{{\mathrm{cot}}^{2}\mathrm{\theta }-1}={\mathrm{tan}}^{2}\mathrm{\theta }$

#### Question 19:

(i) $\frac{\mathrm{tan\theta }}{\left(\mathrm{sec\theta }-1\right)}+\frac{\mathrm{tan\theta }}{\left(\mathrm{sec\theta }+1\right)}=2\mathrm{cosec\theta }$
(ii) $\frac{\mathrm{cot\theta }}{\left(\mathrm{cosec\theta }+1\right)}+\frac{\left(\mathrm{cosec\theta }+1\right)}{\mathrm{cot\theta }}=2\mathrm{sec\theta }$

#### Question 20:

(i) $\frac{\mathrm{sec\theta }-1}{\mathrm{sec\theta }+1}=\frac{{\mathrm{sin}}^{2}\mathrm{\theta }}{\left(1+\mathrm{cos\theta }{\right)}^{2}}$
(ii) $\frac{\mathrm{sec\theta }-\mathrm{tan\theta }}{\mathrm{sec\theta }+\mathrm{tan\theta }}=\frac{{\mathrm{cos}}^{2}\mathrm{\theta }}{\left(1+\mathrm{sin\theta }{\right)}^{2}}$

#### Question 21:

$\frac{\mathrm{sin\theta }}{\left(\mathrm{cot\theta }+\mathrm{cosec\theta }\right)}-\frac{\mathrm{sin\theta }}{\left(\mathrm{cot\theta }-\mathrm{cosec\theta }\right)}=2$

#### Question 22:

(i) $\frac{\mathrm{sin\theta }-\mathrm{cos\theta }}{\mathrm{sin\theta }+\mathrm{cos\theta }}+\frac{\mathrm{sin\theta }+\mathrm{cos\theta }}{\mathrm{sin\theta }-\mathrm{cos\theta }}=\frac{2}{\left(2{\mathrm{sin}}^{2}\mathrm{\theta }-1\right)}$
(ii) $\frac{\mathrm{sin\theta }+\mathrm{cos\theta }}{\mathrm{sin\theta }-\mathrm{cos\theta }}+\frac{\mathrm{sin\theta }-\mathrm{cos\theta }}{\mathrm{sin\theta }+\mathrm{cos\theta }}=\frac{2}{\left(1-2{\mathrm{cos}}^{2}\mathrm{\theta }\right)}$

#### Question 23:

$\frac{{\mathrm{cos}}^{3}\mathrm{\theta }+{\mathrm{sin}}^{3}\mathrm{\theta }}{\mathrm{cos\theta }+\mathrm{sin\theta }}+\frac{{\mathrm{cos}}^{3}\mathrm{\theta }-{\mathrm{sin}}^{3}\mathrm{\theta }}{\mathrm{cos\theta }-\mathrm{sin\theta }}=2$

Hence, LHS= RHS

#### Question 24:

$\left(1-\mathrm{sin\theta }+\mathrm{cos\theta }{\right)}^{2}=2\left(1+\mathrm{cos\theta }\right)\left(1-\mathrm{sin\theta }\right)$

Hence, LHS = RHS

#### Question 25:

$\frac{1+\mathrm{cos\theta }-{\mathrm{sin}}^{2}\mathrm{\theta }}{\mathrm{sin\theta }\left(1+\mathrm{cos\theta }\right)}=\mathrm{cot\theta }$

Hence, L.H.S. = R.H.S.

(i)
(ii)

#### Question 27:

(i) $\frac{1+\mathrm{cos\theta }+\mathrm{sin\theta }}{1+\mathrm{cos\theta }-\mathrm{sin\theta }}=\frac{1+\mathrm{sin\theta }}{\mathrm{cos\theta }}$
(ii) $\frac{\mathrm{sin\theta }+1-\mathrm{cos\theta }}{\mathrm{cos\theta }-1+\mathrm{sin\theta }}=\frac{1+\mathrm{sin\theta }}{\mathrm{cos\theta }}$

#### Question 28:

$\frac{\mathrm{sin\theta }}{\left(\mathrm{sec\theta }+\mathrm{tan\theta }-1\right)}+\frac{\mathrm{cos\theta }}{\left(\mathrm{cosec\theta }+\mathrm{cot\theta }-1\right)}=1$

Hence, LHS = RHS

#### Question 29:

$\frac{\mathrm{sin\theta }+\mathrm{cos\theta }}{\mathrm{sin\theta }-\mathrm{cos\theta }}+\frac{\mathrm{sin\theta }-\mathrm{cos\theta }}{\mathrm{sin\theta }+\mathrm{cos\theta }}=\frac{2}{\left({\mathrm{sin}}^{2}\mathrm{\theta }-{\mathrm{cos}}^{2}\mathrm{\theta }\right)}=\frac{2}{\left(2{\mathrm{sin}}^{2}\mathrm{\theta }-1\right)}$

Hence, LHS = RHS

#### Question 31:

$\left(1+\mathrm{tan\theta }+\mathrm{cot\theta }\right)\left(\mathrm{sin\theta }-\mathrm{cos\theta }\right)=\left(\frac{\mathrm{sec\theta }}{{\mathrm{cosec}}^{2}\mathrm{\theta }}-\frac{\mathrm{cosec\theta }}{{\mathrm{sec}}^{2}\mathrm{\theta }}\right)$

Hence, LHS = RHS

#### Question 32:

$\frac{{\mathrm{cot}}^{2}\mathrm{\theta }\left(\mathrm{sec\theta }-1\right)}{\left(1+\mathrm{sin\theta }\right)}={\mathrm{sec}}^{2}\mathrm{\theta }.\left(\frac{1-\mathrm{sin\theta }}{1+\mathrm{sec\theta }}\right)$

Hence, LHS = RHS

Hence, LHS = RHS

#### Question 37:

Show that none of the following is an identity:
(i) cos2θ + cos θ = 1
(ii) sin2θ + sin θ = 2
(iii) tan2θ + sin θ = cos2θ

#### Question 38:

Show that each of the following is an identity:
(i) $\frac{{\mathrm{tan}}^{2}\mathrm{\theta }}{\left(1+{\mathrm{tan}}^{2}\mathrm{\theta }\right)}={\mathrm{sin}}^{2}\mathrm{\theta }$
(ii) $\frac{\mathrm{cot\theta }+\mathrm{cos\theta }}{\mathrm{cot\theta }-\mathrm{cos\theta }}=\frac{1+\mathrm{sin\theta }}{1-\mathrm{sin\theta }}$

#### Question 1:

If a cos θ + b sin θ = m and a sin θ − b cos θ = n, prove that (m2 + n2) = (a2 + b2).

#### Question 2:

If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that (x2y2) = (a2b2).

#### Question 3:

prove that $\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}\right)=2.$

#### Question 4:

If (sec θ + tan θ) = m and (sec θ − tan θ) = n, show that mn = 1.

#### Question 5:

If (cosec θ + cot θ) = m and (cosec θ − cot θ) = n, show that mn = 1.

#### Question 6:

If x = a cos3θ and y = b sin3θ, prove that ${\left(\frac{x}{a}\right)}^{2/3}+{\left(\frac{y}{b}\right)}^{2/3}=1.$

#### Question 7:

If (tan θ + sin θ) = m and (tan θ − sin θ) = n, prove that (m2 n2)2 = 16mn.

#### Question 8:

If (cot θ + tan θ) = m and (sec θ − cos θ) = n, prove that (m2n)2/3 − (mn2)2/3 = 1.

#### Question 9:

If (cot θ + tan θ) = m and (sec θ − cos θ) = n, prove that (m2n)2/3 − (mn2)2/3 = 1.

#### Question 10:

If (cos θ + sin θ) = 1, prove that (cos θ − sin θ) = ± 1.

#### Question 11:

If tan A = n tan B and sin A = m sin B, prove that cos2A = $\frac{\left({m}^{2}-1\right)}{{n}^{2}-1}.$

#### Question 12:

If (cosec θ − sin θ) = a3 and (sec θ − cos θ) = b3, prove that ${a}^{2}{b}^{2}\left({a}^{2}+{b}^{2}\right)=1.$

#### Question 13:

If a cos3θ + 3a sin2θ cos θ = m and a sin3θ + 3a sin θ cos2θ = n, prove that
(m + n)2/3 + (mn)2/3 = 2a2/3.

#### Question 14:

If (2 sin θ + 3 cos θ) = 2, show that (3 sin θ − 2 cos θ) = ± 3.

#### Question 15:

If sin θ + cos θ = $\sqrt{2}$ sin (90° − θ), show that cot θ = $\left(\sqrt{2}+1\right)$.

#### Question 16:

If cos θ + sin θ = $\sqrt{2}$ sin θ, show that sin θ − cos θ = $\sqrt{2}$ cos θ.

#### Question 17:

If tan θ = $\frac{a}{b}$, show that .

Hence, LHS = RHS

#### Question 1:

If sin A + sin2 A = 1, then (cos2 A + cos4 A) = ?
(a) $\frac{1}{2}$
(b) 1
(c) 2
(d) 3

(b) 1

#### Question 2:

If cos A + cos2 A = 1, then (sin2 A + sin4 A) = ?
(a) 1
(b) 2
(c) 3
(d) 4

(a) 1

#### Question 3:

(sec A + tan A) (1 − sin A) = ?
(a) sin A
(b) cos A
(c) sec A
(d) cosec A

(b) cos A

#### Question 4:

(1 + tan A + sec A)(1 + cot − cosec A) = ?
(a) 1
(b) 0
(c) 2
(d) 4

(c) 2

#### Question 5:

$\frac{\left(1+{\mathrm{tan}}^{2}A\right)}{\left(1+{\mathrm{cot}}^{2}A\right)}=?$
(a) tan2 A
(b) cot2 A
(c) sec2 A
(d) cosec2 A

(a) tan2 A

#### Question 6:

(sin A + cos A)2 + (sin A − cos A)2 = ?
(a) 1
(b) 2
(c) 3
(d) 4

(c) 2

#### Question 7:

(sec4 A − sec2 A) = ?
(a) tan4 A − tan2 A
(b) tan2 A + tan4 A
(c) tan2 A − tan4 A
(d) None of these

(b) tan2 A + tan4 A

$\left({\text{sec}}^{4}A-{\text{sec}}^{2}A\right)\phantom{\rule{0ex}{0ex}}={\text{sec}}^{2}A\left({\text{sec}}^{2}A-1\right)\phantom{\rule{0ex}{0ex}}={\text{sec}}^{2}A\left({\text{tan}}^{2}A\right)\phantom{\rule{0ex}{0ex}}=\left(1+{\text{tan}}^{2}A\right)\left({\text{tan}}^{2}A\right)\phantom{\rule{0ex}{0ex}}={\text{tan}}^{2}A+{\text{tan}}^{4}A\phantom{\rule{0ex}{0ex}}$

#### Question 8:

(cos4θ − sin4θ)=?
(a) 1 − 2 sin2θ
(b) 1 − 2 cos2θ
(c) 2 − sin2θ
(d) 2 − cos2θ

a) 1 − 2 sin2θ

$\left({\text{cos}}^{4}\theta -{\text{sin}}^{4}\theta \right)\phantom{\rule{0ex}{0ex}}={\left({\text{cos}}^{2}\theta \right)}^{2}-{\left({\text{sin}}^{2}\theta \right)}^{2}\phantom{\rule{0ex}{0ex}}=\left({\text{cos}}^{2}\theta +{\text{sin}}^{2}\theta \right)\left({\text{cos}}^{2}\theta -{\text{sin}}^{2}\theta \right)\phantom{\rule{0ex}{0ex}}=1\left(1-{\text{sin}}^{2}\theta -{\text{sin}}^{2}\theta \right)\phantom{\rule{0ex}{0ex}}=1-2{\text{sin}}^{2}\theta \phantom{\rule{0ex}{0ex}}$

#### Question 9:

$\frac{\mathrm{sin\theta }}{\left(1+\mathrm{cos\theta }\right)}=?$
(a) $\frac{\mathrm{cos\theta }}{\left(1-\mathrm{sin\theta }\right)}$
(b) $\frac{\left(1-\mathrm{cos\theta }\right)}{\mathrm{sin\theta }}$
(c) $\frac{\left(1-\mathrm{sin\theta }\right)}{\mathrm{cos\theta }}$
(d) None of these

(b) $\frac{\left(1-\mathrm{cos\theta }\right)}{\mathrm{sin\theta }}$

#### Question 10:

$\frac{\mathrm{sin\theta }}{\left(1+\mathrm{cos\theta }\right)}+\frac{\mathrm{sin\theta }}{\left(1-\mathrm{cos\theta }\right)}=?$
(a) 2 sin θ
(b) 2 cos θ
(c) 2 sec θ
(d) 2 cosec θ

(d) 2 cosec θ

#### Question 11:

$\frac{\mathrm{sin\theta }}{\left(1-\mathrm{cot\theta }\right)}+\frac{\mathrm{cos\theta }}{\left(1-\mathrm{tan\theta }\right)}=?$
(a) (cos θ + sin θ)
(b) (cos θ − sin θ)
(c) 0
(d) 2 tan θ

(a) (cos θ + sin θ)

$\frac{\text{sin}\theta }{\left(1-c\text{ot}\theta \right)}+\frac{\text{cos}\theta }{\left(1-\text{tan}\theta \right)}\phantom{\rule{0ex}{0ex}}\text{=}\frac{\text{sin}\theta }{1-\frac{\text{cos}\theta }{\text{sin}\theta }}+\frac{\text{cos}\theta }{1-\frac{\text{sin}\theta }{\text{cos}\theta }}\phantom{\rule{0ex}{0ex}}=\left(\frac{\text{sin}\theta }{\frac{\text{sin}\theta -c\text{os}\theta }{\text{sin}\theta }}\right)+\left(\frac{\text{cos}\theta }{\frac{\text{cos}\theta -s\text{in}\theta }{\text{cos}\theta }}\right)\phantom{\rule{0ex}{0ex}}\text{=}\left(\frac{{\text{sin}}^{2}\theta }{\text{sin}\theta -c\text{os}\theta }\right)+\left(\frac{{\text{cos}}^{2}\theta }{\text{cos}\theta -s\text{in}\theta }\right)\phantom{\rule{0ex}{0ex}}\text{=}\left(\frac{{\text{cos}}^{2}\theta }{\text{cos}\theta -s\text{in}\theta }\right)-\left(\frac{{\text{sin}}^{2}\theta }{\text{cos}\theta -s\text{in}\theta }\right)\phantom{\rule{0ex}{0ex}}\text{=}\left(\frac{{\text{cos}}^{2}\theta -s{\text{in}}^{2}\theta }{\text{cos}\theta -s\text{in}\theta }\right)\phantom{\rule{0ex}{0ex}}\text{=}\left\{\frac{\left(c\text{os}\theta \text{+sin}\theta \right)\left(c\text{os}\theta -s\text{in}\theta \right)}{\left(c\text{os}\theta -s\text{in}\theta \right)}\right\}\phantom{\rule{0ex}{0ex}}\text{=cos}\theta +s\text{in}\theta \phantom{\rule{0ex}{0ex}}$

#### Question 12:

$\frac{\left(1-{\mathrm{tan}}^{2}\mathrm{\theta }\right)}{\left(1+{\mathrm{tan}}^{2}\mathrm{\theta }\right)}=?$
(a) (sin2θ − cos2θ)
(b) (cos2θ − sin2θ)
(c) (cot2θ − tan2θ)
(d) (tan2θ − cot2θ)

(b) (cos2θ − sin2θ)

#### Question 13:

(a) (sec A + tan A)
(b) (sec A − tan A)
(c) sec A tan A
(d) None of these

(a) (sec A + tan A)

#### Question 14:

(a) (sec A + tan A)
(b) (sec A − tan A)
(c) sec A tan A
(d) None to these

(b) (sec A − tan A)

#### Question 15:

(a) (cosec A − cot A)
(b) (cosec A + cot A)
(c) cosec A cot A
(d) None of these

(a) (cosec A − cot A)

#### Question 16:

(a) (cosec A − cot A)
(b) (cosec A + cot A)
(c) cosec A cot A
(d) None of these

(b) (cosec A + cot A)

#### Question 17:

(a) (sec A − tan A)
(b) (sec A + tan A)
(c) sec A tan A
(d) None of these

(a) (sec A − tan A)

#### Question 18:

(sin4θ − cos4θ + 1) cosec2θ = ?
(a) 4
(b) 3
(c) 2
(d) 1

(c) 2

#### Question 19:

If x = a cos θ and y = b sin θ, then (b2x2 + a2y2) = ?
(a) a2 + b2
(b) a2b2
(c) ab
(d) a4b4

(b) a2b2

#### Question 20:

If x = a sec θ cos ϕ, y = b θ sin ϕ and z = c tan θ, then $\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}\right)=?$
(a) $\left(1+\frac{{z}^{2}}{{c}^{2}}\right)$
(b) $\left(1-\frac{{z}^{2}}{{c}^{2}}\right)$
(c) $\left(\frac{{z}^{2}}{{c}^{2}}-1\right)$
(d) $\frac{{z}^{2}}{{c}^{2}}$

(a) $\left(1+\frac{{z}^{2}}{{c}^{2}}\right)$

#### Question 21:

$\sqrt{\frac{\mathrm{sec\theta }-1}{\mathrm{sec\theta }+1}}+\sqrt{\frac{\mathrm{sec\theta }+1}{\mathrm{sec\theta }-1}}=?$
(a) 2 sin θ
(b) 2 cos θ
(c) 2 cosec θ
(d) 2 sec θ

(c) 2 cosec θ

#### Question 22:

If (sin θ + cos θ) = p and (sec θ + cosec θ) = q, then q(p2 − 1) = ?
(a) 2
(b) 2p
(c) $\frac{1}{{p}^{2}}$
(d) $\frac{1}{p}$

(b) 2p

#### Question 23:

If (cos θ + sin θ) = $\sqrt{2}$ cos θ, then (cos θ − sin θ) = ?
(a) $\sqrt{2}$ sin θ
(b) $\sqrt{2}$ sec θ
(c) $\sqrt{2}$ cosec θ
(d) None of these

(a) $\sqrt{2}$ sin θ

#### Question 24:

$\frac{\left(\mathrm{sin\theta }-2{\mathrm{sin}}^{3}\mathrm{\theta }\right)}{\left(2{\mathrm{cos}}^{3}\mathrm{\theta }-\mathrm{cos\theta }\right)}=?$
(a) tan θ
(b) cot θ
(c) sec θ
(d) cosec θ

(a) tan θ

#### Question 25:

(cosec θ − cot θ)2 = ?
(a) $\frac{1+\mathrm{cos\theta }}{1-\mathrm{cos\theta }}$
(b) $\frac{1-\mathrm{cos\theta }}{1+\mathrm{cos\theta }}$
(c)  $\frac{1+\mathrm{sin\theta }}{1-\mathrm{sin\theta }}$
(d) $\frac{1-\mathrm{sin\theta }}{1+\mathrm{sin\theta }}$

(b) $\frac{1-\mathrm{cos\theta }}{1+\mathrm{cos\theta }}$

${\left(\text{cosec}\theta -\text{cot}\theta \right)}^{2}\phantom{\rule{0ex}{0ex}}\text{=}{\left(\frac{1}{\text{sin}\theta }-\frac{\text{cos}\theta }{\text{sin}\theta }\right)}^{2}\phantom{\rule{0ex}{0ex}}\text{=}{\left(\frac{1-\text{cos}\theta }{\text{sin}\theta }\right)}^{2}\phantom{\rule{0ex}{0ex}}\text{=}\frac{{\left(1-\text{cos}\theta \right)}^{2}}{{\text{sin}}^{2}\theta }\phantom{\rule{0ex}{0ex}}\text{=}\frac{{\left(1-\text{cos}\theta \right)}^{2}}{\left(1-{\text{cos}}^{2}\theta \right)}\phantom{\rule{0ex}{0ex}}\text{=}\frac{{\left(1-\text{cos}\theta \right)}^{2}}{\left(1+\text{cos}\theta \right)\left(1-\text{cos}\theta \right)}\phantom{\rule{0ex}{0ex}}\text{=}\frac{\left(1-\text{cos}\theta \right)}{\left(1+\text{cos}\theta \right)}\phantom{\rule{0ex}{0ex}}$

#### Question 26:

If tan θ = $\frac{a}{b},$ then $\frac{\left(\mathrm{cos\theta }+\mathrm{sin\theta }\right)}{\left(\mathrm{cos\theta }-\mathrm{sin\theta }\right)}=?$
(a) $\frac{a+b}{a-b}$
(b) $\frac{a-b}{a+b}$
(c) $\frac{b+a}{b-a}$
(d) $\frac{b-a}{b+a}$

(c) $\frac{b+a}{b-a}$

#### Question 27:

If (sin θ + cos θ) = $\sqrt{3}$, then (tan θ + cot θ) = ?
(a) $\frac{1}{\sqrt{3}}$
(b) $\frac{1}{2}$
(c) 1
(d) $\frac{3}{2}$

(c) 1

#### Question 28:

If (sin θ + cos θ) = a and (sin3θ + cos3θ) = b, then (3a − 2b) = ?
(a) a3
(b) b3
(c) 0
(d) 1