Question #08ca1

Feb 14, 2017

$L H S = \frac{\sec x \sin x}{\tan x + \cot x}$

$= \frac{\sec x \sin x}{\sin \frac{x}{\cos} x + \cos \frac{x}{\sin} x}$

$= \frac{\sec x \sin x}{\frac{{\sin}^{2} x + {\cos}^{2} x}{\cos x \sin x}}$

$= \sec x \cdot \sin x \cdot \cos x \cdot \sin x$

$= {\sin}^{2} x$

Feb 14, 2017

see explanation.

Explanation:

Making use of the following $\textcolor{b l u e}{\text{trigonometric identities}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\tan x = \sin \frac{x}{\cos} x , \cot x = \cos \frac{x}{\sin} x} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{and } \textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\sec x = \frac{1}{\cos} x} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{left side } = \frac{\sec x \times \sin x}{\tan x + \cot x}$

$\textcolor{w h i t e}{\times \times \times} = \frac{\sin \frac{x}{\cos} x}{\sin \frac{x}{\cos} x + \cos \frac{x}{\sin} x}$

$\textcolor{w h i t e}{\times \times \times} = \frac{\sin \frac{x}{\cos} x}{\frac{{\sin}^{2} x + {\cos}^{2} x}{\sin x \cos x}}$

$\textcolor{w h i t e}{\times \times \times} = \sin \frac{x}{\cancel{\cos x}} \times \frac{\sin x \cancel{\cos x}}{1}$

$\textcolor{w h i t e}{\times \times \times} = {\sin}^{2} x = \text{right side"rArr" verified}$