Question

sinA(1+tanA) + cosa(1+cotA) = secA +cosecA

Answer 1

To prove sin A(1+ tan A)+ cos A(1 + cot A) = sec A + cosec A.

LHS = sin A(1+ tan A)+ cos A(1 + cot A)

= sin A + sin^2 A/ cos A + cos A + cos^2 A/ sin A

= sin A + cos A + [sin^3 A + cos^3 A]/sin A cos A

=[ sin^2 A cos A + cos^2 A sin A + sin^3 A + cos^3 A]/sin A cos A

= [ sin^2 A cos A +cos^3 A + cos^2 A sin A + sin^3 A]/sin A cos A

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

= [cos A +sin A]/sin A cos A

= (1/sin A) + (1/cos A)

= cosec A + sec A = RHS.

Proved.

Answer 2

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