Question

cos theta/1+sin theta = 1-sin theta/cos theta​

Answer 1

Step-by-step explanation:

cos a/1+sin a=1-sin a/cos a

cross multiply

cos²a=(1+sin a)(1-sina)

cos²a=1-sin²a

cos²a=cos²a

Answer 2

To prove that

$\sf{ \frac{cos \: x}{1 + sin \: x} = \frac{1 - sin \: x}{cos \: x} }$

Regrouping the terms,

$\sf{ \frac{cos \: x}{1 + sin \: x} - \frac{1 - \: sin \: x}{cos \: x} = 0 }$

LHS:

$\sf{ \frac{cos \: x}{1 + \: sin \: x} - \frac{1 - sin \: x}{cos \: x} } \\ \\ = \sf{\frac{cos {}^{2} - (1 - sin {}^{2}x) }{(1 + \: sin \: x)cos \: x}} \\ \\ = \sf{\frac{cos {}^{2}x - cos {}^{2}x }{(1 + sin \: x)cos \: x} } \\ \\ = 0$

Thus,LHS=RHS

Hence,proved