Question

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

Answer 1

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

$=\frac{\cot \theta -1+cosec\theta }{\cot \theta +1-cosec\theta }=\frac{\cot \theta -(cose{c}^2\theta -{\cot }^2\theta )+cosec\theta }{\cot \theta +1-cosec\theta }$

$=\frac{\cot \theta -\left[\right(cosec\theta -\cot \theta \left)\right(cosec\theta +\cot \theta \left)\right]+cosec\theta }{\cot \theta +1-cosec\theta }$

$=\frac{(\cot \theta +cosec\theta )[1-cosec\theta +\cot \theta ]}{\cot \theta +1-cosec\theta }$

$=\cot \theta +cosec\theta$.