= sec A ( – sin A)(sec A + tan A) = ( sin A) ( sin A)( + sin A) = RHS Example : Prove that cot A – cos A cosec A – cot A + cos A cosec A + Solution : LHS = cot A – cos A cot A + cos A cosec A – cosec A + = RHS Example : Prove that sin cos , sin cos using the identity sec = + tan . Solution : Since we will apply the identity involving sec and tan , let us first convert the LHS (of the identity we need to prove) in terms of sec and tan by dividing numerator and denominator by cos LHS = sin – cos + sin + cos – = (tan {(tan } (tan (tan {(tan } (tan (tan ) (tan {tan } (tan – (tan ) (tan – , which is the RHS of the identity, we are required to prove. EXERCISE . .
Express the trigonometric ratios sin A, sec A and tan A in terms of cot A. . Write all the other trigonometric ratios of A in terms of sec A. .
Choose the correct option. Justify your choice. (i) sec A – tan A = (A) (B) (C) (D) (ii) ( + tan + sec ) ( + cot – cosec ) = (A) (B) (C) (D) – (iii) (sec A + tan A) ( – sin A) = (A) sec A (B) sin A (C) cosec A (D) cos A (iv) A + cot A (A) sec A (B) – (C) cot A (D) tan A . Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(i) (cosec – cot ) = cos cos (ii) sec A