Exercise 8.3
1. Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.
Solution :
click here to watch on youtube
2. Write all the other trigonometric ratios of ∠A in terms of sec A.
Solution :
click here to watch on youtube
3. Choose the correct option. Justify your choice.
(i) 9 sec2A – 9 tan2A =
(A) 1 (B) 9 (C) 8 (D) 0
(ii) (1 + tan θ + sec θ) (1 + cot θ – cosec θ)
(A) 0 (B) 1 (C) 2 (D) – 1
Solution :
click here to watch on youtube
(iii) (sec A + tan A) (1 – sin A) =
(A) sec A (B) sin A (C) cosec A (D) cos A
(iv) 1+tan2A/1+cot2A =
(A) sec2 A (B) -1 (C) cot2A (D) tan2A
Solution :
click here to watch on youtube
4. Prove the following identities, where the angles involved are acute angles for which the
expressions are defined.
(i) (cosec θ – cot θ)2 = (1-cos θ)/(1+cos θ)
Solution :
click here to watch on youtube
(ii) cos A/(1+sin A) + (1+sin A)/cos A = 2 sec A
Solution :
click here to watch on youtube
(iii) tan θ/(1-cot θ) + cot θ/(1-tan θ) = 1 + sec θ cosec θ
[Hint : Write the expression in terms of sin θ and cos θ]
Solution :
click here to watch on youtube
(iv) (1 + sec A)/sec A = sin2A/(1-cos A)
[Hint : Simplify LHS and RHS separately]
Solution :
click here to watch on youtube
(v) ( cos A–sin A+1)/( cos A +sin A–1) = cosec A + cot A, using the identity cosec2A = 1+cot2A.
Solution :
click here to watch on youtube
Solution :
click here to watch on youtube
(vii) (sin θ – 2sin3θ)/(2cos3θ-cos θ) = tan θ
Solution :
click here to watch on youtube
(viii) (sin A + cosec A)2 + (cos A + sec A)2 = 7+tan2A+cot2A
Solution :
click here to watch on youtube
(ix) (cosec A – sin A)(sec A – cos A) = 1/(tan A+cotA)
[Hint : Simplify LHS and RHS separately]
Solution :
click here to watch on youtube
(x) (1+tan2A/1+cot2A) = (1-tan A/1-cot A)2 = tan2A
Solution :