Question

Simplifying Trigonometric Expressions-

Write the trigonometric expression $\mathit{t}\mathit{a}{\mathit{n}}^{\mathbf{2}}\mathit{x}\mathbf{-}\mathit{s}\mathit{e}{\mathit{c}}^{\mathbf{2}}\mathit{x}$in terms of sine and cosine, and then simplify.

Short answer

On simplifying expression we get $\mathbf{-}\mathbf{1}$

Step-by-step solution

Step 1. Given information

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

Step 2. Concept used

Write the expression in sine and cosine terms first. Then for simplification solve it until it converts into simplest form.

Step 3. Calculation

Now, write ${\tan }^{2}x-{\sec }^{2}x$in form of $sineand\cos ine$:

$\begin{array}{l}{\tan }^{2}x={\left(\frac{\sin x}{\cos x}\right)}^2=\frac{\sin x}{{\cos }^{2}x}\\ {\sec }^{2}x={\left(\frac{1}{\cos x}\right)}^2=\frac{1}{{\cos }^{2}x}\\ \therefore {\tan }^{2}x-{\sec }^{2}x\\ =\frac{{\sin }^{2}x}{{\cos }^{2}x}-\frac{1}{{\cos }^{2}x}\\ =\frac{{\sin }^{2}x-1}{{\cos }^{2}x}\\ =\frac{-\left(1-{\sin }^{2}x\right)}{{\cos }^{2}x}\\ =\frac{-{\cos }^{2}x}{{\cos }^{2}x}\\ =-1\end{array}$