00950 6_Aa) Prove that SinA (1+ TanA) + cos A (1+ CotA) =SecA + Cosec

1.	00950 6_Aa) Prove that SinA (1+ TanA) + cos A (1+ CotA) =SecA + Cosec A (3)
Answer» 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.

00950 6_Aa) Prove that SinA (1+ TanA) + cos A (1+ CotA) =SecA + Cosec A (3)

Answer»

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.