Question

Establish each identity: csc Θ/ 1+ csc Θ = 1-sin Θ/ cos^2 Θ

Answer 1

$sin^2(\theta)+cos^2(\theta)=1\implies cos^2(\theta)=1-sin^2(\theta) \\\\[-0.35em] \rule{34em}{0.25pt}$

$\cfrac{csc(\theta )}{1+csc(\theta )}=\cfrac{1-sin(\theta )}{cos^2(\theta )} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{csc(\theta )}{1+csc(\theta )}\implies \cfrac{~~\frac{1}{sin(\theta )} ~~}{1+\frac{1}{sin(\theta )}}\implies \cfrac{~~\frac{1}{sin(\theta )} ~~}{\frac{sin(\theta )+1}{sin(\theta )}}\implies \cfrac{1}{sin(\theta )}\cdot \cfrac{sin(\theta )}{sin(\theta )+1}$

$\cfrac{1}{sin(\theta )+1}\implies \cfrac{1}{1+sin(\theta )}\implies \stackrel{\textit{multiplying by the conjugate of the denominator}}{\underset{\textit{difference of squares}}{\cfrac{1}{1+sin(\theta )}\cdot \cfrac{1-sin(\theta )}{1-sin(\theta )}}} \\\\\\ \cfrac{1-sin(\theta )}{1^2-sin^2(\theta )}\implies \cfrac{1-sin(\theta )}{1-sin^2(\theta )}\implies \cfrac{1-sin(\theta )}{cos^2(\theta )}$