Question

1. \\(\frac{sin x(\tan x + cot x)}{cos x}\\)

Explanation:

Step1: Substitute trigonometric identities

We know that \(\tan x=\frac{\sin x}{\cos x}\) and \(\cot x = \frac{\cos x}{\sin x}\).
Substitute these into the expression: \(\frac{\sin x(\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x})}{\cos x}\).

Step2: Simplify the expression inside the parentheses

Find a common denominator for \(\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}\). The common denominator is \(\sin x\cos x\).
\(\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}=\frac{\sin^{2}x+\cos^{2}x}{\sin x\cos x}\).
Since \(\sin^{2}x+\cos^{2}x = 1\), the expression becomes \(\frac{1}{\sin x\cos x}\).

Step3: Multiply and simplify

Now we have \(\frac{\sin x\times\frac{1}{\sin x\cos x}}{\cos x}\).
The \(\sin x\) in the numerator and denominator cancels out. We get \(\frac{1}{\cos^{2}x}\).
Since \(\frac{1}{\cos^{2}x}=\sec^{2}x\).

Answer:

\(\sec^{2}x\)