5. Prove that Sin0(1+tan0) +cos0(1+cot0)-Sece + cosece

1.	5. Prove that Sin0(1+tan0) +cos0(1+cot0)-Sece + cosece
Answer» Let theta = 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. Hence proved

Answer»

Let theta = 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.

Hence proved

Discussion