Question

Prove that: $\frac{sin\theta }{1-cos\theta }+\frac{tan\theta }{cos\theta +1}=sec\theta .cosec\theta +cot\theta$

Answer 1

LHS

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

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

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

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

$=\frac{sin\theta co{s}^2\theta +sin\theta }{cos\theta si{n}^2\theta }$

$=\frac{sin\theta co{s}^2\theta }{cos\theta si{n}^2\theta }+\frac{sin\theta }{cos\theta si{n}^2\theta }$

$=\frac{cos\theta }{sin\theta }+\frac{1}{sin\theta cos\theta }=cot\theta +cosec\theta sec\theta$ = RHS