Question

prove the following....

please answer guys....​

Answer 1

Step-by-step explanation:

LHS = $\frac{1}{sec\alpha + tan\alpha } - \frac{1}{cos\alpha }$

= $\frac{1}{\frac{1}{cos\theta } + \frac{sin\theta }{cos\theta } } - \frac{1}{cos\theta }$

= $\frac{cos\theta }{1 + sin\theta } - \frac{1}{cos\theta }$

= $\frac{cos\theta (1 - sin\theta)}{(1 + sin\theta)(1 - sin\theta)} - \frac{1}{cos\theta}$

= $\frac{cos\theta (1 - sin\theta)}{1 - sin^{2}\theta } - \frac{1}{cos\theta}$

= $\frac{cos\theta (1 - sin\theta)}{cos^{2}\theta } - \frac{1}{cos\theta}$

= $\frac{cos\theta (1 - sin\theta) - cos\theta}{cos^{2}\theta }$

= $\frac{- sin\theta}{cos\theta}$ = $(- tan\theta)$

RHS = $\frac{1}{cos\theta } - \frac{1}{sec\theta - tan\theta }$

= $\frac{1}{cos\theta} - \frac{cos\theta}{1 - sin\theta }$

= $\frac{cos\theta - cos\theta(1 + sin\theta)}{cos^{2}\theta }$

= $\frac{- sin\theta}{cos\theta}$ = $(- tan\theta)$

∴ LHS = RHS

Hence Proved

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