Problem 62 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\frac{1+\sec ^{2} x}{1+\tan ^{2} x}=1+\cos ^{2} x$$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \(1 + \cos^2 x\).

Step 1: Rewrite using Pythagorean Identity

First, we recall the Pythagorean identity: \(1 + \tan^2 x = \sec^2 x\). This allows us to rewrite the expression \(\frac{1+\sec^2 x}{1+\tan^2 x}\) as \(\frac{1+\sec^2 x}{\sec^2 x}\).

Step 2: Simplify the Fraction

Next, simplify the expression \(\frac{1+\sec^2 x}{\sec^2 x}\). This can be written as \(\frac{1}{\sec^2 x} + 1\).

Step 3: Use Reciprocal Identity

Remember that \(\sec^2 x = \frac{1}{\cos^2 x}\). Thus, \(\frac{1}{\sec^2 x} = \cos^2 x\). Replace \(\frac{1}{\sec^2 x}\) in the expression to get \(\cos^2 x + 1\).

Step 4: Compare with the Right Side

Now, our expression is \(\cos^2 x + 1\), which is identical to the right side of the identity \(1 + \cos^2 x\). Thus, we have verified the identity.

Key Concepts

Pythagorean identity

In trigonometry, the Pythagorean identity is incredibly useful for simplifying expressions and verifying identities. This identity is derived from the Pythagorean theorem applied to a unit circle, and it states that:

• \(1 + \tan^2 x = \sec^2 x\)

This particular form of the Pythagorean identity is especially useful when dealing with expressions that involve \(\tan\) and \(\sec\). By recognizing and substituting these into their identity form, you simplify complex trigonometric expressions.
In the given exercise, we use this identity to transform the denominator \(1 + \tan^2 x\) into \(\sec^2 x\), which directly simplifies our fraction. This crucial step shows the power of the Pythagorean identity in reducing the complexity of trigonometric problems.

reciprocal identities

Reciprocal identities in trigonometry are relationships that define one trigonometric function in terms of another. They are particularly useful in simplifying expressions and solving equations. The reciprocal identities you should know for this exercise are:

• \(\sec x = \frac{1}{\cos x}\)
• Thus, \(\sec^2 x = \frac{1}{\cos^2 x}\)

Recognizing these identities enables us to express terms involving \(\sec\) in terms of \(\cos\).
In our exercise, after simplifying the fraction to \(\frac{1}{\sec^2 x} + 1\), we utilize the reciprocal identity for \(\sec^2 x\). This allows us to express \(\frac{1}{\sec^2 x}\) as \(\cos^2 x\), simplifying the expression to match the identity on the right side.
Understanding these reciprocal relationships is a fundamental skill in trigonometry that makes tackling such exercises much easier.

simplifying expressions

Simplifying trigonometric expressions involves breaking them down into simpler, equivalent forms. This often includes using identities such as Pythagorean or reciprocal identities, to transform terms in a way that reduces complexity.
In our exercise, the expression \(\frac{1+\sec^2 x}{1+\tan^2 x}\) was simplified by applying the Pythagorean identity, changing the fraction to \(\frac{1+\sec^2 x}{\sec^2 x}\). Then, simplifying this fraction further to \(\frac{1}{\sec^2 x} + 1\), enabled another application of reciprocal identities.
Ultimately, simplifying led us to \(\cos^2 x + 1\), which conveniently matched the target identity \(1 + \cos^2 x\).
Remember, the key to simplifying expressions is to identify known identities and relationships within the terms, letting those guide the manipulations and resulting simplifications.