Question

Simplifying Trigonometric Expressions-

Simplify the trigonometric expression$\frac{\mathbf{s}\mathbf{e}{\mathbf{c}}^{\mathbf{2}}\mathbf{x}\mathbf{-}\mathbf{1}}{\mathbf{s}\mathbf{e}{\mathbf{c}}^{\mathbf{2}}\mathbf{x}}$

Short answer

On simplifying expression we get $\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{x}$

Step-by-step solution

Step 1. Given information

A trigonometric expression as$\frac{{\sec }^{2}x-1}{{\sec }^{2}x}$

Step 2. Concept used

Apply trigonometric identities to simplify the problem below. Simplify it as much as can. The simplest form is the correct answer.

Step 3. Calculation

Now, simplify $\frac{{\sec }^{2}x-1}{{\sec }^{2}x}$using identities:

Assume$\frac{{\sec }^{2}x-1}{{\sec }^{2}x}\dots \dots Eq\left(1\right)$

Now divide by ${\sec }^{2}x$both numerators

$\begin{array}{l}\frac{{\sec }^{2}x-1}{{\sec }^{2}x}=1-\frac{1}{{\sec }^{2}x}\\ \frac{{\sec }^{2}x-1}{{\sec }^{2}x}=1-{\cos }^{2}x\end{array}$

Now use formula $\left(1-{\cos }^2\alpha \right)={\sin }^2\alpha$

$\frac{{\sec }^{2}x-1}{{\sec }^{2}x}={\sin }^{2}x$