Question

If $\theta$ and $\varphi$ are complementary angles then what is the value of $\text{cosec}\theta \sec \varphi -\cot \theta \tan \varphi$

Answer 1

Given that, $\theta$ and $\varphi$ are complementary angles,

Hence, $\theta +\varphi =90°$

$\Rightarrow \theta =90°-\varphi$

$\therefore \csc \theta \sec \varphi -\cot \theta \tan \varphi =\csc (90°-\varphi )\sec \varphi -\cot (90°-\varphi )\tan \varphi$

$\Rightarrow \sec \varphi \sec \varphi -\tan \varphi \tan \varphi$

$\Rightarrow {\sec }^2\varphi -{\tan }^2\varphi$

$=1\{\because {\sec }^2\theta -{\tan }^2\theta =1\}$

Hence, if $\theta$ and $\varphi$ are complementary angles, then $\csc \theta \sec \varphi -\cot \theta \tan \varphi =1$