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2. Write all the other trigonometric ratios of ∠A in terms of sec A.
3. Evaluate :
(i) (sin
2
63° + sin
2
27°)/(cos
2
17° + cos
2
73°)
(ii) sin 25° cos 65° + cos 25° sin 65°
(ii) sin 25° cos 65° + cos 25° sin 65°
(ii) (1 + tan θ + sec θ) (1 + cot θ - cosec θ)
(A) 0 (B) 1 (C) 2 (D) - 1
(iii) (secA + tanA) (1 - sinA) =
(A) secA (B) sinA (C) cosecA (D) cosA
(iv) 1+tan
2
A/1+cot
2
A =
(A) sec
2
A (B) -1 (C) cot
2
A (D) tan
2
A
5. 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)
(ii) cos A/(1+sin A) + (1+sin A)/cos A = 2 sec A
(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]
(iv) (1 + sec A)/sec A = sin
2
A/(1-cos A)
[Hint : Simplify LHS and RHS separately]
(v) (cos A–sin A+1)/(cos A+sin A–1) = cosec A + cot A,using the identity cosec
2
A = 1+cot
2
A.
(vi) √1 + sin A/1 - sin A = sec A+ tan A
(vii) (sin θ - 2sin
3
θ)/(2cos
3
θ-cos θ) = tan θ
(viii) (sin A + cosec A)
2
+ (cos A + sec A)
2
= 7+tan
2
A+cot
2
A
(ix) (cosec A – sin A)(sec A – cos A) = 1/(tan A+cotA)
[Hint : Simplify LHS and RHS separately]
(x) (1+tan
2
A/1+cot
2
A) = (1-tan A/1-cot A)
2
= tan
2
A
(ii) cos A/(1+sin A) + (1+sin A)/cos A = 2 sec A
(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]
(iv) (1 + sec A)/sec A = sin
2
A/(1-cos A)
[Hint : Simplify LHS and RHS separately]
(v) (cos A–sin A+1)/(cos A+sin A–1) = cosec A + cot A,using the identity cosec
2
A = 1+cot
2
A
(viii) (sin A + cosec A)
2
+ (cos A + sec A)
2
= 7+tan
2
A+cot
2
A
(ix) (cosec A – sin A)(sec A – cos A) = 1/(tan A+cotA)
[Hint : Simplify LHS and RHS separately]
(x) (1+tan
2
A/1+cot
2
A) = (1-tan A/1-cot A)
2
= tan
2
A
