Question

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

(sin A + cosec A)² + (cos A + sec A)² = 7 + tan² A + cot² A

Answer 1

(sin A + cosec A)² + (cos A + sec A)² = 7 + tan² A + cot² A

L.H.S = (sin A + cosec A)² + (cos A + sec A)²

= sin²A + cosec²A + 2sinA cosecA + cos²A + sec²A + 2cosA secA

= (sin²A + cos²A) + (cosec²A + sec²A) + 2sinA`(1/sinA)`+ 2cosA`(1/cosA)`

= (1) + (1 + cot²A + 1 + tan²A) + (2) + (2)

= 7 + tan²A + cot²A

= R.H.S