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Chapter 5: Problem 45

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer. $$\frac{1}{\tan ^{2} x+1}$$

Short Answer

Expert verified
The simplified form of the expression \( \frac{1}{\tan^2 x + 1} \) is \( \cos^2 x \).

Step by step solution

01

Identify the Fundamental Identity

Recognize the fundamental trigonometric identity known as the Pythagorean Identity, which is \( \tan^2 x + 1 = \sec^2 x \). This can be used to simplify the expression.
02

Apply the Fundamental Identity

Replace \( \tan^2 x + 1 \) in the denominator of the expression with \( \sec^2 x \). So, the expression now becomes \( \frac{1}{\sec^2 x} \).
03

Simplify the Expression

The reciprocal of \( \sec x \) is \( \cos x \). Therefore, \( \frac{1}{\sec^2 x} \) simplifies to \( \cos^2 x \).

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Most popular questions from this chapter

(a) use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval \([0,2 \pi),\) and (b) solve the trigonometric equation and demonstrate that its solutions are the \(x\) -coordinates of the maximum and minimum points of \(f .\) (Calculus is required to find the trigonometric equation.) Function $$f(x)=\sin ^{2} x+\cos x$$ Trigonometric Equation $$2 \sin x \cos x-\sin x=0$$

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{11 \pi}{12}=\frac{3 \pi}{4}+\frac{\pi}{6}$$

Explain in your own words how knowledge of algebra is important when solving trigonometric equations.

Use the Quadratic Formula to solve the equation in the interval \([0,2 \pi)\). Then use a graphing utility to approximate the angle \(x\). $$3 \tan ^{2} x+4 \tan x-4=0$$

Solve the multiple-angle equation. $$\cos \frac{x}{2}=\frac{\sqrt{2}}{2}$$
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