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Question

Evaluate :

$$\cos \left \{\tan^{-1} \left (-\dfrac {3}{4}\right )\right \}$$


Solution

$$\cos{\left\{{\tan}^{-1}{\left(\dfrac{-3}{4}\right)}\right\}}$$

Let $${\tan}^{-1}{\left(\dfrac{-3}{4}\right)}=a$$

$$\Rightarrow\,\tan{a}=\dfrac{-3}{4}$$

$$\Rightarrow\,{\tan}^{2}{a}=\dfrac{9}{16}$$

$$\Rightarrow\,{\sec}^{2}{a}=1+{\tan}^{2}{a}=1+\dfrac{9}{16}=\dfrac{16+9}{16}=\dfrac{25}{16}$$

$$\Rightarrow \sec{a}=\dfrac{5}{4}$$

$$\Rightarrow\,\cos{a}=\dfrac{1}{\sec{a}}=\dfrac{1}{\dfrac{5}{4}}=\dfrac{4}{5}$$

Now, $$\cos{\left\{{\tan}^{-1}{\left(\dfrac{-3}{4}\right)}\right\}}=\cos{a}=\dfrac{4}{5}$$                                                   $$\because ba={\tan}^{-1}{\left(\dfrac{-3}{4}\right)}$$

Mathematics

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