Question

Without using trigonometric tables, evaluate:
$\text{cosec}{}^2{57}^{\circ }-{\tan }^2{33}^{\circ }+\cos {44}^{\circ }\text{cosec}{46}^{\circ }$

Answer 1

Relation between trigonometric ratios of complementary angles are,

$\begin{array}{cc}\sin \left(90-\theta \right)& =\cos \theta \\ \text{cosec}\left(90-\theta \right)& =\sec \theta \\ \tan \left(90-\theta \right)& =\cot \theta \end{array}$

By using above relations,

$\begin{array}{cc}{cosec}^2{57}^{\circ }-{\tan }^2{33}^{\circ }+\cos {44}^{\circ }cosec{46}^{\circ }& ={cosec}^2({90}^{\circ }-{33}^{\circ })-{\tan }^2{33}^{\circ }+\cos ({90}^{\circ }-{46}^{\circ })cosec{46}^{\circ }\\ & ={\sec }^2{33}^{\circ }-{\tan }^2{33}^{\circ }+\sin {46}^{\circ }\times \frac{1}{\sin {46}^{\circ }}\\ & =1+1\dots [\text{By using,}{\sec }^2\theta -{\tan }^2\theta =1]\\ & =2\end{array}$