Question

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

Answer 1

$\begin{array}{c}Giventhat,\\ \frac{\cos \theta -\sin \theta +1}{\cos \theta +\sin \theta -1}=\cos ec\theta +\cot \theta \\ Now,\\ Divideboth{N}_{r}and{D}_{r}with\sin \theta \\ =\frac{\frac{\cos \theta -\sin \theta +1}{\sin \theta }}{\frac{\cos \theta +\sin \theta -1}{\sin \theta }}\\ =\frac{\cot \theta -1+\cos ec\theta }{\cot \theta +1-\cos ec\theta }\\ =\frac{\cot \theta +\cos ec\theta -\left(\cos e{c}^2\theta -{\cot }^2\theta \right)}{\cot \theta -\cos ec\theta +1}\\ =\frac{\cot \theta +\cos ec\theta -\cos e{c}^2\theta +{\cot }^2\theta }{\cot \theta -\cos ec\theta +1}\\ =\frac{\cot \theta +\cos ec\theta \left(1-\left(\cos e{c}^2\theta -{\cot }^2\theta \right)\right)}{\cot \theta -\cos ec\theta +1}\\ =\cot \theta +\cos ec\theta \\ HenceL.H.S=R.H.S\end{array}$