Question

If $\cos \theta =\frac{3}{5},\pi <\theta <\frac{3\pi }{2}$ find the value of $\frac{\csc \theta +\cot \theta }{\sec \theta -\tan \theta }$

Answer 1

Consider the following question.

$\cos \theta =\frac{3}{5},\pi <\theta <\frac{3\pi }{2}$

Find the value of $\frac{\csc \theta +\cot \theta }{\sec \theta -\tan \theta }$

Using a Pythagoras theorem, we get.

$A{C}^2=A{B}^2+B{C}^2$

${\left(5\right)}^2=A{B}^2+{\left(3\right)}^2$

$AB=4$

$\csc \theta =\frac{5}{4},\sec \theta =\frac{5}{3},\tan \theta =\frac{4}{3},\cot \theta =\frac{3}{4}$

$\frac{\csc \theta +\cot \theta }{\sec \theta -\tan \theta }=\frac{\frac{5}{4}+\frac{3}{4}}{\frac{5}{3}-\frac{4}{3}}$

$=6$

Hence, this is the required answer.