Question

if cosec theta +cot theta =k,then prove that cos theta=k2-1/k2+1​

Answer 1

Step-by-step explanation:

We have,

$\cosec( \theta) + \cot(\theta) = k$

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

$\implies\frac{2 \cos^{2} ( \frac{\theta }{2} ) }{2 \sin( \frac{\theta}{2} ) \cos( \frac{\theta}{2} ) } = k$

$\implies \cot( \frac{\theta}{2} ) = k$

$\implies \tan( \frac{\theta}{2} ) = \frac{1}{k}$

Now,

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

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

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