Question

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

Answer 1

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

$=$dividing Nr and Dr by $sin\theta$ we get,

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

$=$$\frac{cot\theta -1+cosec\theta }{cot\theta +1-cosec\theta }$

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

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

$=$$cot\theta +cosec\theta$$=$RHS