Question

question 9
\frac{\sec^{2}x - 1}{\sec^{2}x}

Explanation:

Step1: Use trigonometric identity

We know that \(1 + \tan^{2}x=\sec^{2}x\), so \(\sec^{2}x - 1=\tan^{2}x\).
The expression becomes \(\frac{\tan^{2}x}{\sec^{2}x}\).

Step2: Express in sine and cosine

Since \(\tan x=\frac{\sin x}{\cos x}\) and \(\sec x=\frac{1}{\cos x}\), then \(\frac{\tan^{2}x}{\sec^{2}x}=\frac{\frac{\sin^{2}x}{\cos^{2}x}}{\frac{1}{\cos^{2}x}}\).

Step3: Simplify the expression

\(\frac{\frac{\sin^{2}x}{\cos^{2}x}}{\frac{1}{\cos^{2}x}}=\sin^{2}x\)

Answer:

\(\sin^{2}x\)