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Chapter 5: Problem 33

Verify each identity. $$\sec ^{2} x \csc ^{2} x=\sec ^{2} x+\csc ^{2} x$$

Short Answer

Expert verified
The given trigonometric identity \(\sec ^{2} x \csc ^{2} x=\sec ^{2} x+\csc ^{2} x\) is verified and holds true for all values of \(x\).

Step by step solution

01

Break Down the Left Side of the Identity

Convert the left side of the identity from secant and cosecant form to cosine and sine form respectively: \(\sec ^{2} x \csc ^{2} x = (\frac{1}{\cos ^{2} x})(\frac{1}{\sin^{2}x})=\frac{1}{\cos ^{2} x \sin^{2}x}\).
02

Simplify the Left Side of the Identity

Apply the Pythagorean identity to simplify: \(\frac{1}{\cos ^{2} x \sin ^{2} x}=\frac{1}{(\sin ^{2} x + \cos ^{2} x)^{2}}= \frac{1}{1}= 1\).
03

Break Down and Simplify the Right Side of the Identity

Convert the right side of the identity from secant and cosecant form to cosine and sine form respectively: \(\sec ^{2} x+\csc ^{2} x = \frac{1}{\cos ^{2} x} + \frac{1}{\sin^{2}x} = \frac{\sin ^{2} x + \cos ^{2} x}{\cos^{2}x \sin^{2}x}= \frac{1}{\cos^{2}x \sin^{2}x}= \frac{1}{1}= 1\). Since both the sides of the equation are equal, the given identity is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Secant Function
The secant function, often abbreviated as sec, is a fundamental trigonometric function. It is closely linked with the cosine function.

The relationship between secant and cosine can be expressed by the formula:
  • \( \sec x = \frac{1}{\cos x} \)

Therefore, the secant value at any angle is simply one divided by the cosine value at that angle.

While the cosine function gives us the ratio of the adjacent side over the hypotenuse in a right triangle, the secant function gives us the ratio of the hypotenuse over the adjacent side.

Understanding the secant function is essential in trigonometry because it helps us solve different types of trigonometric identities and equations swiftly.

Let's take a deeper look at another related function: the cosecant function.
Cosecant Function
The cosecant function, abbreviated as csc, is another primary trigonometric function that complements the sine function. The formula that defines this relationship is:
  • \( \csc x = \frac{1}{\sin x} \)

Just as secant is the reciprocal of cosine, cosecant is the reciprocal of sine.

When you want to determine the cosecant value of an angle, you take the reciprocal of the sine of that angle.

In a right triangle, while the sine represents the ratio of the opposite side over the hypotenuse, the cosecant is the ratio of the hypotenuse over the opposite side.

Just like the secant function, the cosecant function is vital in solving various trigonometric problems, especially those involving trigonometric identities.

Now, let's explore how these functions interact with one of the pivotal identities in trigonometry: the Pythagorean identity.
Pythagorean Identity
The Pythagorean identity is one of the cornerstone equations in trigonometry. This identity provides a straightforward relationship between sine and cosine:
  • \( \sin^2 x + \cos^2 x = 1 \)

This identity is incredibly useful as it allows us to simplify complex trigonometric expressions.

In the context of the given exercise, the Pythagorean identity was key to simplifying the original equation.

By relating the secant and cosecant functions back to sine and cosine (using their reciprocal relationships), the equation becomes straightforward to solve because \( \sin^2 x + \cos^2 x = 1 \).

Mastering this identity not only assists in simplifying trigonometric identities but also equips students with a useful tool for tackling a wide array of mathematical problems.

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Most popular questions from this chapter

In Exercises \(85-96,\) use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$\cos ^{2} x-\cos x-1=0$$

In Exercises \(85-96,\) use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$\tan ^{2} x-3 \tan x+1=0$$

In Exercises \(63-84,\) use an identity to solve each equation on the interval \([0,2 \pi)\) $$\sin \left(x+\frac{\pi}{4}\right)+\sin \left(x-\frac{\pi}{4}\right)=1$$

In Exercises \(97-116,\) use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$\cos ^{2} x+5 \cos x-1=0$$

In Exercises \(121-126,\) solve each equation on the interval \([0,2 \pi)\) $$10 \cos ^{2} x+3 \sin x-9=0$$
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