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#### Question 1:

If and , where 0 < A, $\mathrm{B}<\frac{\mathrm{\pi }}{2}$, find the values of the following:

(i) sin (A + B)
(ii) cos (A + B)
(iii) sin (A − B)
(iv) cos (A − B)

Now,

#### Question 2:

(a) If , where $\frac{\mathrm{\pi }}{2}$< A < π and 0 < B < $\frac{\mathrm{\pi }}{2}$, find the following:

(i) sin (A + B)
(ii) cos (A + B)

(b) If , where A and B both lie in second quadrant, find the value of sin (A + B).

#### Question 3:

If , where π < A < $\frac{3\mathrm{\pi }}{2}\mathrm{and}\frac{3\mathrm{\pi }}{2}$< B < 2π, find the following:

(i) sin (A + B)
(ii) cos (A + B)

#### Question 4:

If , where π < A < $\frac{3\mathrm{\pi }}{2}$and 0 < B < $\frac{\mathrm{\pi }}{2}$, find tan (A + B).

#### Question 5:

If , where $\frac{\mathrm{\pi }}{2}$< A < π and $\frac{3\mathrm{\pi }}{2}$ < B < 2π, find tan (AB).

#### Question 6:

If , where $\frac{\mathrm{\pi }}{2}$ < A < π and 0 < B < $\frac{\mathrm{\pi }}{2}$, find the following:

(i) tan (A + B)
(ii) tan (AB)

#### Question 7:

Evaluate the following:

(i) sin 78° cos 18° − cos 78° sin 18°
(ii) cos 47° cos 13° − sin 47° sin 13°
(iii) sin 36° cos 9° + cos 36° sin 9°
(iv) cos 80° cos 20° + sin 80° sin 20°

#### Question 8:

If , where A lies in the second quadrant and B in the third quadrant, find the values of the following:

(i) sin (A + B)
(ii) cos (A + B)
(iii) tan (A + B)

#### Question 9:

Prove that: $\frac{7\pi }{12}+\mathrm{cos}\frac{\pi }{12}=\mathrm{sin}\frac{5\pi }{12}-\mathrm{sin}\frac{\pi }{12}$

LHS = cos105o + cos15o
= cos(90o + 15o) + cos(90o$-$ 75o)
= - sin 15o + sin 75o                      [As cos(90o+A) = $-$ sin A and cos(90o$-$B) = sin B]
= sin 75o$-$ sin 15o
= RHS
Hence proved.

Prove that .

Prove that
(i) .

(ii)
(ii)

(i)

#### Question 12:

Prove that:
(i) $\mathrm{sin}\left(\frac{\pi }{3}-x\right)\mathrm{cos}\left(\frac{\pi }{6}+x\right)+\mathrm{cos}\left(\frac{\pi }{3}-x\right)\mathrm{sin}\left(\frac{\pi }{6}+x\right)=1$
(ii) $\mathrm{sin}\left(\frac{4\mathrm{\pi }}{9}+7\right)\mathrm{cos}\left(\frac{\mathrm{\pi }}{9}+7\right)-\mathrm{cos}\left(\frac{4\mathrm{\pi }}{9}+7\right)\mathrm{sin}\left(\frac{\mathrm{\pi }}{9}+7\right)=\frac{\sqrt{3}}{2}$
(iii) $\mathrm{sin}\left(\frac{3\mathrm{\pi }}{8}-5\right)\mathrm{cos}\left(\frac{\mathrm{\pi }}{8}+5\right)+\mathrm{cos}\left(\frac{3\mathrm{\pi }}{8}-5\right)\mathrm{sin}\left(\frac{\mathrm{\pi }}{8}+5\right)=1$

(i)

(ii)

(iii)

Prove that .

#### Question 14:

(i) If , prove that $A+B=\frac{\mathrm{\pi }}{4}$.
(ii) If , then prove that $A-B=\frac{\mathrm{\pi }}{4}$.

(i)

(ii)

#### Question 15:

Prove that:
(i) ${\mathrm{cos}}^{2}45°-{\mathrm{sin}}^{2}15°=\frac{\sqrt{3}}{4}$
(ii) sin2 (n + 1) A − sin2nA = sin (2n + 1) A sin A.

(i)

Hence proved.

(ii)

#### Question 16:

Prove that:

(i)
(ii)
(iii)
(iv) sin2B = sin2A + sin2 (AB) − 2 sin A cos B sin (A B)
(v) cos2A + cos2B − 2 cos A cos B cos (A + B) = sin2 (A + B)
(vi)

#### Question 17:

Prove that:
(i) tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x
(ii) $\mathrm{tan}\frac{\pi }{12}+\mathrm{tan}\frac{\pi }{6}+\mathrm{tan}\frac{\pi }{12}\mathrm{tan}\frac{\pi }{6}=1$
(iii) tan 36° + tan 9° + tan 36° tan 9° = 1
(iv) tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x

Prove that:

#### Question 19:

Prove that sin2 (n + 1) A − sin2nA = sin (2n + 1) A sin A.

#### Question 20:

If tan A = x tan B, prove that .

#### Question 21:

If tan (A + B) = x and tan (AB) = y, find the values of tan 2A and tan 2B.

#### Question 22:

If cos A + sin B = m and sin A + cos B = n, prove that 2 sin (A + B) = m2 + n2 − 2.

#### Question 23:

If tan A + tan B = a and cot A + cot B = b, prove that cot (A + B) $\frac{1}{a}-\frac{1}{b}$.

Given:

#### Question 24:

If x lies in the first quadrant and , then prove that:

#### Question 25:

If tan x + , then prove that .

#### Question 26:

If sin (α + β) = 1 and sin (α − β)$=\frac{1}{2}$, where 0 ≤ α, $\mathrm{\beta }\le \frac{\mathrm{\pi }}{2}$, then find the values of tan (α + 2β) and tan (2α + β).

#### Question 27:

If α, β are two different values of x lying between 0 and 2π, which satisfy the equation 6 cos x + 8 sin x = 9, find the value of sin (α + β).

#### Question 28:

If sin α + sin β = a and cos α + cos β = b, show that

(i)
(ii)

(i)

Now,

From (1) and (2), we have

(ii)

Prove that:

(i)
(ii)
(iii)

#### Question 30:

If sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.

Given:

#### Question 31:

If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x2.

#### Question 32:

If angle $\theta$ is divided into two parts such that the tangents of one part is $\lambda$ times the tangent of other, and $\varphi$ is their difference, then show that $\mathrm{sin}\theta =\frac{\lambda +1}{\lambda -1}\mathrm{sin}\varphi$.                                                                                                                                                          [NCERT EXEMPLER]

Let $\alpha$ and $\beta$ be the two parts of angle $\theta$. Then,

$\theta =\alpha +\beta$ and $\varphi =\alpha -\beta$               (Given)

Now,

Applying componendo and dividendo, we get

#### Question 33:

If $\mathrm{tan}\theta =\frac{\mathrm{sin}\alpha -\mathrm{cos}\alpha }{\mathrm{sin}\alpha +\mathrm{cos}\alpha }$, then show that $\mathrm{sin}\alpha +\mathrm{cos}\alpha =\sqrt{2}\mathrm{cos}\theta$.                              [NCERT EXEMPLER]

$\mathrm{tan}\theta =\frac{\mathrm{sin}\alpha -\mathrm{cos}\alpha }{\mathrm{sin}\alpha +\mathrm{cos}\alpha }$

Dividing numerator and denominator on the RHS by $\mathrm{cos}\alpha$, we get

Now,

$\mathrm{sin}\alpha +\mathrm{cos}\alpha \phantom{\rule{0ex}{0ex}}=\mathrm{sin}\left(\frac{\mathrm{\pi }}{4}+\theta \right)+\mathrm{cos}\left(\frac{\mathrm{\pi }}{4}+\theta \right)\phantom{\rule{0ex}{0ex}}=\mathrm{sin}\frac{\mathrm{\pi }}{4}\mathrm{cos}\theta +\mathrm{cos}\frac{\mathrm{\pi }}{4}\mathrm{sin}\theta +\mathrm{cos}\frac{\mathrm{\pi }}{4}\mathrm{cos}\theta -\mathrm{sin}\frac{\mathrm{\pi }}{4}\mathrm{sin}\theta \phantom{\rule{0ex}{0ex}}=\frac{1}{\sqrt{2}}\mathrm{cos}\theta +\frac{1}{\sqrt{2}}\mathrm{sin}\theta +\frac{1}{\sqrt{2}}\mathrm{cos}\theta -\frac{1}{\sqrt{2}}\mathrm{sin}\theta \phantom{\rule{0ex}{0ex}}=\frac{2}{\sqrt{2}}\mathrm{cos}\theta \phantom{\rule{0ex}{0ex}}=\sqrt{2}\mathrm{cos}\theta$

#### Question 34:

If α and β are two solutions of the equation a tan x + b sec x = c, then find the values of sin (α + β) and cos (α + β).

#### Question 1:

Find the maximum and minimum values of each of the following trigonometrical expressions:
(i) 12 sin x − 5 cos x
(ii) 12 cos x + 5 sin x + 4
(iii)
(iv) sin x − cos x + 1

(i)

(ii)

(iii)

(iv)

#### Question 2:

Reduce each of the following expressions to the sine and cosine of a single expression:
(i)
(ii) cos x − sin x
(iii) 24 cos x + 7 sin x

#### Question 3:

Show that sin 100° − sin 10° is positive.

#### Question 4:

Prove that lies between .

#### Question 1:

If α + β − γ = π and sin2 α +sin2 β − sin2 γ = λ sin α sin β cos γ, then write the value of λ.

#### Question 2:

If x cos θ = y cos , then write the value of $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$.

#### Question 3:

Write the maximum and minimum values of 3 cos x + 4 sin x + 5.

#### Question 4:

Write the maximum value of 12 sin x − 9 sin2x.

#### Question 5:

If 12 sin x − 9sin2x attains its maximum value at x = α, then write the value of sin α.

#### Question 6:

Write the interval in which the value of 5 cos x + 3 cos $\left(x+\frac{\pi }{3}\right)+3$ lies.

#### Question 7:

If tan (A + B) = p and tan (AB) = q, then write the value of tan 2B.

#### Question 8:

If , then write the value of tan x tan y.

#### Question 9:

If a = b , then write the value of ab + bc + ca.

#### Question 10:

If A + B = C, then write the value of tan A tan B tan C.

#### Question 11:

If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).

#### Question 12:

If tan $\mathrm{\alpha }=\frac{1}{1+{2}^{-x}}$and , then write the value of α + β lying in the interval .

#### Question 1:

The value of ${\mathrm{sin}}^{2}\frac{5\mathrm{\pi }}{12}-{\mathrm{sin}}^{2}\frac{\mathrm{\pi }}{12}$ is
(a) $\frac{1}{2}$
(b) $\frac{\sqrt{3}}{2}$
(c) 1
(d) 0

(b)  $\frac{\sqrt{3}}{2}$

#### Question 2:

If A + B + C = π, then sec A (cos B cos C − sin B sin C) is equal to
(a) 0
(b) −1
(c) 1
(d) None of these

(b)  −1
π = 180°
$\mathrm{sec}A\left(\mathrm{cos}B\mathrm{cos}C-\mathrm{sin}B\mathrm{sin}C\right)=\frac{\mathrm{cos}B\mathrm{cos}\left(\mathrm{\pi }-\left(A+B\right)\right)-\mathrm{sin}B\mathrm{sin}\left(\mathrm{\pi }-\left(A+B\right)\right)}{\mathrm{cos}A}$

We know that, ,

$\therefore \mathrm{sec}A\left(\mathrm{cos}B\mathrm{cos}C-\mathrm{sin}B\mathrm{sin}C\right)=\frac{\mathrm{cos}B\mathrm{cos}\left(A+B\right)-\mathrm{sin}B\mathrm{sin}\left(A+B\right)}{\mathrm{cos}A}$

Now, using the identities $\mathrm{cos}\left(A+B\right)=\mathrm{cos}A\mathrm{cos}B-\mathrm{sin}A\mathrm{sin}B$ and $\mathrm{sin}\left(A+B\right)=\mathrm{sin}A\mathrm{cos}B+\mathrm{cos}A\mathrm{sin}B$, we get

$\mathrm{sec}A\left(\mathrm{cos}B\mathrm{cos}C-\mathrm{sin}B\mathrm{sin}C\right)=\frac{-\mathrm{cos}A\mathrm{cos}{B}^{2}+\mathrm{cos}B\mathrm{sin}A\mathrm{sin}B-\mathrm{sin}B\mathrm{sin}A\mathrm{cos}B-{\mathrm{sin}}^{2}B\mathrm{cos}A}{\mathrm{cos}A}$

$⇒\mathrm{sec}A\left(\mathrm{cos}B\mathrm{cos}C-\mathrm{sin}B\mathrm{sin}C\right)=\frac{-\mathrm{cos}A\left({\mathrm{cos}}^{2}B+{\mathrm{sin}}^{2}B\right)}{\mathrm{cos}A}\phantom{\rule{0ex}{0ex}}⇒\mathrm{sec}A\left(\mathrm{cos}B\mathrm{cos}C-\mathrm{sin}B\mathrm{sin}C\right)=\frac{-\mathrm{cos}A}{\mathrm{cos}A}=-1$

#### Question 3:

tan 20° + tan 40° + $\sqrt{3}$ tan 20° tan 40° is equal to

(a) $\frac{\sqrt{3}}{4}$
(b) $\frac{\sqrt{3}}{2}$
(c) $\sqrt{3}$
(d) 1

(c) $\sqrt{3}$

#### Question 4:

If , then the value of A + B is

(a) 0
(b) $\frac{\mathrm{\pi }}{2}$
(c) $\frac{\mathrm{\pi }}{3}$
(d) $\frac{\mathrm{\pi }}{4}$

(d)  $\frac{\mathrm{\pi }}{4}$

#### Question 5:

If 3 sin x + 4 cos x = 5, then 4 sin x − 3 cos x =
(a) 0
(b) 5
(c) 1
(d) None of these

(a) 0

#### Question 6:

If in âˆ†ABC, tan A + tan B + tan C = 6, then cot A cot B cot C =

(a) 6
(b) 1
(c) $\frac{1}{6}$
(d) None of these

(c) $\frac{1}{6}$
In triangle ABC,

If tan A+tan B+tan C =6,
tan A tan B tan C =6

#### Question 7:

tan 3A − tan 2A − tan A =

(a) tan 3 A tan 2 A tan A
(b) −tan 3 A tan 2 A tan A
(c) tan A tan 2 A − tan 2 A tan 3 A − tan 3 A tan A
(d) None of these

(a)

#### Question 8:

If A + B + C = π, then is equal to
(a) tan A tan B tan C
(b) 0
(c) 1
(d) None of these

(c) 1
π = 180°
Using tan(180 – A) = -tan A, we get:

#### Question 9:

If , where P and Q both are acute angles. Then, the value of PQ is
(a) $\frac{\mathrm{\pi }}{6}$

(b) $\frac{\mathrm{\pi }}{3}$

(c) $\frac{\mathrm{\pi }}{4}$

(d) $\frac{\mathrm{\pi }}{12}$

(b) 60â° = $\frac{\mathrm{\pi }}{3}$

$=\frac{1}{7}×\frac{13}{14}+\frac{4\sqrt{3}}{7}×\frac{3\sqrt{3}}{14}\phantom{\rule{0ex}{0ex}}=\frac{13+36}{98}\phantom{\rule{0ex}{0ex}}$
$=\frac{49}{98}$

Hence, the correct answer is option B.

#### Question 10:

If cot (α + β) = 0, sin (α + 2β) is equal to

(a) sin α
(b) cos 2 β
(c) cos α
(d) sin 2 α

(a)

(a) tan 55°
(b) cot 55°
(c) −tan 35°
(d) −cot 35°

(a)

#### Question 12:

The value of is
(a)
(b) 0
(c)
(d) $\frac{1}{2}$

(a)

#### Question 13:

If tan θ1 tan θ2 = k, then

(a) $\frac{1+k}{1-k}$
(b) $\frac{1-k}{1+k}$
(c) $\frac{k+1}{k-1}$
(d) $\frac{k-1}{k+1}$

(a) $\frac{1+k}{1-k}$

$\frac{1+\mathrm{tan}{\theta }_{1}\mathrm{tan}{\theta }_{2}}{1-\mathrm{tan}{\theta }_{1}\mathrm{tan}{\theta }_{2}}\phantom{\rule{0ex}{0ex}}=\frac{1+k}{1-k}$

#### Question 14:

If sin (π cos x) = cos (π sin x), then sin 2 x =

(a) $±\frac{3}{4}$

(b) $±\frac{4}{3}$

(c) $±\frac{1}{3}$

(d) none of these

#### Question 15:

If $\mathrm{tan}\theta =\frac{1}{2}$ and $\mathrm{tan}\varphi =\frac{1}{3}$, then the value of $\theta +\varphi$ is

(a) $\frac{\mathrm{\pi }}{6}$                               (b) $\mathrm{\pi }$                               (c) 0                               (d) $\frac{\mathrm{\pi }}{4}$

It is given that $\mathrm{tan}\theta =\frac{1}{2}$ and $\mathrm{tan}\varphi =\frac{1}{3}$.

Now,

Hence, the correct answer is option D.

#### Question 16:

The value of cos (36° − A) cos (36° + A) + cos (54° + A) cos (54° − A) is

(a) sin 2A
(b) cos 2A
(c) cos 3A
(d) sin 3A

(b)  cos 2A

#### Question 17:

If tan (π/4 + x) + tan (π/4 − x) = a, then tan2 (π/4 + x) + tan2 (π/4 − x) =
(a) a2 + 1
(b) a2 + 2
(c) a2 − 2
(d) None of these

(c) ${a}^{2}-2$

#### Question 18:

If tan (AB) = 1 and sec (A + B) = $\frac{2}{\sqrt{3}}$, the smallest positive value of B is

(a)
(b)
(c) $\frac{13\mathrm{\pi }}{24}$
(d)

(b)

#### Question 19:

If AB = π/4, then (1 + tan A) (1 − tan B) is equal to

(a) 2
(b) 1
(c) 0
(d) 3

(a)  $2$

#### Question 20:

The maximum value of ${\mathrm{sin}}^{2}\left(\frac{2\pi }{3}+x\right)+{\mathrm{sin}}^{2}\left(\frac{2\pi }{3}-x\right)$ is
(a) 1/2
(b) 3/2
(c) 1/4
(d) 3/4

(b) $\frac{3}{2}$
$\frac{2\pi }{3}=120°$

#### Question 21:

If cos (AB)$=\frac{3}{5}$and tan A tan B = 2, then

(a)
(b)
(c)
(d)

(a) $\frac{1}{5}$

#### Question 22:

If tan 69° + tan 66° − tan 69° tan 66° = 2k, then k =

(a) −1
(b) $\frac{1}{2}$
(c) $-\frac{1}{2}$
(d) None of these

(c)$\frac{-1}{2}$

#### Question 23:

If $\mathrm{tan}\alpha =\frac{x}{x+1}$ and $\mathrm{tan}\beta =\frac{1}{2x+1}$, then $\alpha +\beta$ is equal to

(a) $\frac{\mathrm{\pi }}{2}$                               (b) $\frac{\mathrm{\pi }}{3}$                               (c) $\frac{\mathrm{\pi }}{6}$                              (d) $\frac{\mathrm{\pi }}{4}$

It is given that $\mathrm{tan}\alpha =\frac{x}{x+1}$ and $\mathrm{tan}\beta =\frac{1}{2x+1}$.
$=\frac{2{x}^{2}+2x+1}{2{x}^{2}+2x+1}\phantom{\rule{0ex}{0ex}}=1$