Prove: cos A/ (1+ sin A ) + (1+ sin A) / cos A = 2 sec A

To prove this, we will use trigonometric identities.

Answer: Hence Proved cos A / (1 + sin A) + (1 + sin A) / cos A = 2 sec A

See detailed solution.

Explanation: 

We will start from LHS, cos A / (1 + sin A) + (1 + sin A) / cos A

cos A / (1 + sin A) + (1 + sin A) / cos A = [cos²A + (1 + sin A)²] / [cos A (1 + sin A)]                                                       

= [cos²A + 1² + sin²A + 2 sin A] / [cos A (1 + sin A)]

= [1 + 1 + 2 sin A] / [cos A (1 + sin A)]                                                            

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

= 2 / cos A

= 2 sec A

= RHS

Hence proved that cos A / ( 1 + sin A ) + ( 1 + sin A) / cos A = 2 sec A

Thus, cos A / ( 1 + sin A ) + (1 + sin A ) / cos A = 2 sec A