Question

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

Answer 1

Given: $\frac{\cot \theta +\cos ec\theta -1}{\cot \theta -\cos ec\theta +1}=\cot \theta +cosec\theta$

LHS:$\frac{\cot \theta +\cos ec\theta -1}{\cot \theta -\cos ec\theta +1}$

$=\frac{\cot \theta +\cos ec\theta -(\cos e{c}^2\theta -{\cot }^2\theta )}{\cot \theta -\cos ec\theta +1}$

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

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

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

$=\cot \theta +\cos ec\theta$

Hence, proved.