Question

Please help me fast................

Answer 1

Prove

Prove that [tan A + cosec A]2 - [cot B - sec A]2 = 2 tan A cot B ( cosec A + sec B)

Sol:

[tan A + cosec B]2 -[cot B - sec A]2 = 2 tan A cot B [cosec A + sec B]

LHS = [tan2 A + cosec2 B + 2 tan A cosec B] - [cot2 B + sec2 A - 2 cot B sec A]

= [tan2 A + cosec2 B + 2 tan A cosec B - cot2 B - sec2A + 2 cot B sec A]

= [(tan2 A - sec2 A) + (cosec2 B - cot2 B) + 2 sin A / cos A (1 / sin B) + 2 cos B / sin B  (1 / cos A)

= (-1) + (1) + 2(sin A + cos B)/cos A sin B

= 2(sin A + cos B)/cos A sin B

RHS = 2 tan A cot B (cosec A + sec B)

       = 2 sin A/cos A x cos B/sin B (1/sin A + 1/cos B)

       = 2 sin A cos B/cos A sin B [(cos B + sin A) / sin A cos B]
 
       = 2(sin A + cos B)/cos A sin B

∴ LHS = RHS