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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.
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