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#### Question 1:

Without using trigonometric tables, evaluate:
(i)
(ii)
(iii)
(iv)
(v)
(vi)

#### Question 2:

Without using trigonometric tables, prove that:
(i) cos 81° − sin 9° = 0
(ii) tan 71° − cot 19° = 0
(iii) cosec 80° − sec 10° = 0
(iv) cosec272° − tan218° = 1
(v) cos275° + cos215° = 1
(vi) tan266° − cot224° = 0
(vii) sin248° + sin242° = 1
(viii) cos257° − sin233° = 0
(ix) (sin 65° + cos 25°)(sin 65° − cos 25°) = 0

#### Question 3:

Without using trigonometric tables, prove that:

(i) sin53° cos37° + cos53° sin37° = 1
(ii) cos54° cos36° − sin54° sin36° = 0
(iii) sec70° sin20° + cos20° cosec70° = 2
(iv) sin35° sin55° − cos35° cos55° = 0
(v) (sin72° + cos18°)(sin72° − cos18°) = 0
(vi) tan48° tan23° tan42° tan67° = 1

Prove that:

#### Question 5:

Prove that:

(i)
(ii) $\frac{\mathrm{sin\theta }}{\mathrm{cos}\left(90°-\mathrm{\theta }\right)}+\frac{\mathrm{cos\theta }}{\mathrm{sin}\left(90°-\mathrm{\theta }\right)}=2$
(iii)
(iv) $\frac{\mathrm{cos}\left(90°-\mathrm{\theta }\right)\mathrm{sec}\left(90°-\mathrm{\theta }\right)\mathrm{tan\theta }}{\mathrm{cosec}\left(90°-\mathrm{\theta }\right)\mathrm{sin}\left(90°-\mathrm{\theta }\right)\mathrm{cot}\left(90°-\mathrm{\theta }\right)}+\frac{\mathrm{tan}\left(90°-\mathrm{\theta }\right)}{\mathrm{cot\theta }}=2$
(v) $\frac{\mathrm{cos}\left(90°-\mathrm{\theta }\right)}{1+\mathrm{sin}\left(90°-\mathrm{\theta }\right)}+\frac{1+\mathrm{sin}\left(90°-\mathrm{\theta }\right)}{\mathrm{cos}\left(90°-\mathrm{\theta }\right)}=2\mathrm{cosec\theta }$
(vi)
(vii)

#### Question 6:

Prove that:

(i) tan5° tan25° tan30° tan65° tan85° = $\frac{1}{\sqrt{3}}$
(ii) cot12° cot38° cot52° cot60° cot78° = $\frac{1}{\sqrt{3}}$
(iii) cos15° cos35° cosec55° cos60° cosec75° = $\frac{1}{2}$
(iv) cos1° cos2° cos3° ... cos180° = 0
(v) ${\left(\frac{\mathrm{sin}49°}{\mathrm{cos}41°}\right)}^{2}+{\left(\frac{\mathrm{cos}41°}{\mathrm{sin}49°}\right)}^{2}=2$

Disclaimer: The RHS of (v) given in textbook is incorrect. There should be 2 instead 1. The same has been corrected in the solution here.

#### Question 7:

Prove that
(i) sin (70° + θ) − cos (20° − θ) = 0
(ii) tan (55° − θ) − cot (35° + θ) = 0
(iii) cosec (67° + θ) − sec (23° − θ) = 0
(iv) cosec (65 °+ θ)  sec  (25° −  θ) − tan (55° − θ) + cot (35° + θ) = 0
(v) sin (50° + θ ) − cos (40° − θ) + tan 1° tan 10° tan 80° tan 89° = 1.

#### Question 8:

Express each of the following in terms of trigonometric ratios of angles lying between 0° and 45°.

(i) sin67° + cos75°
(ii) cot65° + tan49°
(iii) sec78° + cosec56°
(iv) cosec54° + sin72°

#### Question 9:

If A, B and C are the angles of a $∆$ABC, prove that $\mathrm{tan}\left(\frac{\mathrm{C}+\mathrm{A}}{2}\right)=\mathrm{cot}\frac{\mathrm{B}}{2}$.

#### Question 10:

Hence, the value of $\theta$ is $15°.$

#### Question 11:

If sec2A = cosec(A $-$ 42$°$), where 2A is an acute angle, then find the value of A.       [CBSE2008]

We have,

Hence, the value of A is $44°$.

#### Question 12:

If sin 3 A = cos (A − 26°), where 3 A is an acute angle, find the value of A.

#### Question 13:

If tan 2 A = cot (A − 12°), where 2 A is an acute angle, find the value of A.

#### Question 14:

If sec 4 A = cosec (A − 15°), where 4 A is an acute angle, find the value of A.

Prove that: