Problem 62 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The identity \( \frac{1+\sec ^{2} x}{1+\tan ^{2} x}=1+\cos ^{2} x \) is verified as true.

Step 1: Recall Trigonometric Identities

To verify the identity, we begin by recalling some fundamental trigonometric identities: \( \sec^2 x = 1 + \tan^2 x \) and \( \tan^2 x = \sec^2 x - 1 \). These will be useful in simplifying the left side of the equation.

Step 2: Simplify the Denominator

Using the identity \( \tan^2 x = \sec^2 x - 1 \), we can rewrite the denominator \( 1 + \tan^2 x \) as \( \sec^2 x \). Therefore, the expression becomes:\[ \frac{1 + \sec^2 x}{\sec^2 x} \]

Step 3: Simplify the Fraction

Now simplify the fraction:\[ \frac{1 + \sec^2 x}{\sec^2 x} = \frac{1}{\sec^2 x} + 1 \].Recall that \( \frac{1}{\sec^2 x} = \cos^2 x \). Thus, the expression simplifies to:\[ 1 + \cos^2 x \], which matches the right side of the original equation.

Step 4: Confirm Identity

Both sides of the equation are equal. We have shown that:\[ \frac{1 + \sec^2 x}{1 + \tan^2 x} = 1 + \cos^2 x \] is indeed a valid identity.

Key Concepts

Secant Function

The secant function, often denoted as \( \sec x \), is a crucial part of trigonometry. It is defined as the reciprocal of the cosine function. This means that \( \sec x = \frac{1}{\cos x} \). The secant function is helpful when dealing with right triangles and circles, as it describes the ratio of the hypotenuse to the adjacent side in a right triangle.

Another important identity related to the secant function is \( \sec^2 x = 1 + \tan^2 x \). This equation plays a key role in verifying trigonometric identities, as it connects the secant and tangent functions.

Understanding this identity allows us to simplify expressions that involve both secant and tangent. In the given exercise, using the identity to rewrite \( \tan^2 x \) helps reduce the complexity of the equation and confirms the given trigonometric identity.

Tangent Function

The tangent function, represented as \( \tan x \), is another fundamental trigonometric function. It is defined as the ratio of the sine of an angle to the cosine of that angle. In formula format, this is expressed as \( \tan x = \frac{\sin x}{\cos x} \).

This function appears frequently in trigonometric identities and equations because of its relationship with both sine and cosine. In the identity \( \sec^2 x = 1 + \tan^2 x \), we can see the connection between the secant and tangent functions.

In solving the exercise, we utilized the expression \( \tan^2 x = \sec^2 x - 1 \) to simplify the identity. This relationship simplifies calculations and transforms complex trigonometric expressions into more manageable forms.

Cosine Function

Cosine, denoted as \( \cos x \), is a core component of the trigonometric world. It describes the ratio of the adjacent side to the hypotenuse in a right triangle. As a building block of many trigonometric identities, cosine is often paired with sine to form fundamental relationships like the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \).

In the context of verifying trigonometric identities, cosine frequently appears due to its reciprocal relationship with the secant function. When verifying \( \frac{1+\sec ^{2} x}{1+\tan ^{2} x} = 1+\cos ^{2} x \), you can see cosine's role as \( \frac{1}{\sec^2 x} \) simplifies to \( \cos^2 x \). This connection allows us to transform and simplify trigonometric expressions accurately.

• It simplifies the left-hand side of the equation.
• Helps in rewriting expressions based on known identities.
• Essential for confirming that both sides of an identity are equivalent.

By recognizing the relationships between secant, tangent, and cosine, you can better understand complex trigonometric expressions and make them easier to work with.