Question

If $\tan \theta =-1$ then find the value of $\frac{\sec \theta -\cos ec\theta }{\cos \theta -\sin \theta },$ $\frac{3\pi }{2}<\theta <2\pi$

• A. 2
• B. 1
• C. $-\sqrt{2}$
• D. $\sqrt{2}$

Answer 1

The correct option is A 2

$\frac{\sec \theta -\csc \theta }{\cos \theta -\sin \theta }$

Given $\theta$ lies in fourth quadrant.

$\tan \theta =-1$

$\Rightarrow \sec \theta =\sqrt{2}$

$\Rightarrow \cos \theta =\frac{1}{\sqrt{2}}$

$\Rightarrow \csc \theta =-\sqrt{2}$

$\Rightarrow \sin \theta =-\frac{1}{\sqrt{2}}$

$\frac{\sec \theta -\csc \theta }{\cos \theta -\sin \theta }$

$=\frac{\sqrt{2}+\sqrt{2}}{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}$

$=\frac{2\sqrt{2}\times \sqrt{2}}{2}=2$