Question

prove that lhs =rhs for cos theta by 1+sin theta = 1- sin theta by cos theta

Answer 1

Answer:

Given :

$\displaystyle{\dfrac{\cos\theta}{1+\sin\theta}=\dfrac{1-\sin\theta}{\cos\theta}}$

$\display \text{L.H.S. =\dfrac{\cos\theta}{1+\sin\theta} }\\\\\\\display \text{Multiply and divide by \cos\theta }\\\\\display \text{L.H.S. =\dfrac{\cos\theta}{1+\sin\theta}\times\dfrac{\cos\theta}{\cos\theta} }\\\\\\\display \implies\dfrac{\cos^2\theta}{(1+\sin\theta)\cos\theta}\\\\\\\display \text{We can write \cos^2\theta=1-\sin^2\theta }\\\\\display \implies\dfrac{1-\sin^2\theta}{(1+\sin\theta)\cos\theta}$

$\display \implies\dfrac{1^2-\sin^2\theta}{(1+\sin\theta)\cos\theta}\\\\\\\display \text{Using identity x^2-y^2=(x+y)(x-y) }\\\\\display \implies\dfrac{(1+\sin\theta)(1-\sin\theta)}{(1+\sin\theta)\cos\theta}\\\\\\\display \text{ (1+\sin\theta) cancel out}\\\\\display \implies\dfrac{1-\sin\theta}{\cos\theta}\\\\\\\display \text{L.H.S. = R.H.S.}\\\\\large \text{Hence Proved .}$