Question

show that 1/1-cos theta- 1/1+cos theta=2cosecsquare theta?

Answer 1

We have to prove that :

$\frac{1}{1 - \cos\alpha } +\frac{1}{1 + \cos \alpha } = 2 { \csc}^{2} \alpha$

On taking LHS :

$\frac{1}{1 - \cos \alpha } + \frac{1}{1 + \cos\alpha } \\ \\ = > \frac{1 + \cos \alpha + 1 - \cos\alpha }{(1 - \cos \alpha )(1 + \cos \alpha ) } \\ \\ = > \frac{1 + \cos \alpha + 1 - \cos\alpha }{1 - { \cos }^{2} \alpha } \\ \\ = > \frac{2 }{ { \sin}^{2} \alpha } \\ \\ \: As \: we \: know \: that :\: { \sin }^{2} \alpha + { \cos}^{2} \alpha = 1 \\ \\ = > 2 { \csc }^{2} \alpha = RHS \\ \\ HENCE \: PROVED$