Question

If sec theta = 3/2 find 2cos²theta + 2cot²theta -9

Answer 1

Answer:

The answer is $-\frac{293}{45}$

Step-by-step explanation:

Given $\displaystyle \sec\theta=\frac{3}{2}$

$\displaystyle \Rightarrow \cos^2\theta=\frac{1}{\sec^2\theta}=\frac{4}{9}$

Also we know  $\sec^2\theta-\tan^2\theta=1$

                  $\displaystyle \Rightarrow \frac{9}{4}-\tan^2\theta=1\\\Rightarrow \tan^2\theta=\frac{9}{4}-1=\frac{5}{4}$

$\displaystyle \Rightarrow \cot^2\theta=\frac{1}{\tan^2\theta}=\frac{4}{5}$

Therefore  $2\cos^2\theta+2\cot^2\theta-9$

                $\displaystyle =2\times \frac{4}{9}+2\times \frac{4}{5}-9$

                 $\displaystyle =\frac{40+72}{45}-9=\frac{112}{45}-9=\frac{112-405}{45}=-\frac{293}{45}$