Question

Establish the identity

$\frac{csc\theta +cot\theta }{sec\theta +\tan \theta }=\frac{sec\theta -\tan \theta }{csc\theta -cot\theta }$

Short answer

Rationalizing the numerator and denominator by multiply numerator and denominator by

$$\begin{gathered} csc\theta -cot\theta andsec\theta -\tan \theta \\ weget\frac{csc\theta +cot\theta }{sec\theta +\tan \theta }=\frac{sec\theta -\tan \theta }{csc\theta -cot\theta } \end{gathered}$$

Step-by-step solution

Step 1. Given information

$\frac{csc\theta +cot\theta }{sec\theta +\tan \theta }=\frac{sec\theta -\tan \theta }{csc\theta -cot\theta }$

Step 2. Rationalizing the numerator and denominator

$$\begin{gathered} Wehavegiven, \\ \frac{csc\theta +cot\theta }{sec\theta +\tan \theta }=\frac{sec\theta -\tan \theta }{csc\theta -cot\theta } \\ TakeLHS\frac{csc\theta +cot\theta }{sec\theta +\tan \theta } \end{gathered}$$

Rationalizing the numerator and denominator by multiply numerator and denominator by$csc\theta -cot\theta andsec\theta -\tan \theta$

$$\begin{gathered} \left(\frac{csc\theta +cot\theta }{sec\theta +\tan \theta }\right)\left(\frac{csc\theta -cot\theta }{csc\theta -cot\theta }\right)\left(\frac{sec\theta -\tan \theta }{sec\theta -\tan \theta }\right) \\ =\frac{(cs{c}^2\theta -co{t}^2\theta )(sec\theta -\tan \theta )}{(se{c}^2\theta -{\tan }^2\theta )(csc\theta -cot\theta )}ascs{c}^2\theta -co{t}^2\theta =1 \\ se{c}^2\theta -{\tan }^2\theta =1 \\ =\frac{sec\theta -\tan \theta }{csc\theta -cot\theta } \end{gathered}$$