Problem 68 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\csc ^{2} x-\cot ^{2} x}{\sec ^{2} x}=\cos ^{2} x $$

Step-by-Step Solution

Answer

The identity is verified: LHS simplifies to \(\cos^2 x\), which equals the RHS.

Step 1: Convert Terms to Sine and Cosine

To verify the identity, start by rewriting both the numerator and the denominator of the left-hand side (LHS) in terms of sine and cosine. Recall that \( \csc x = \frac{1}{\sin x} \) and \( \cot x = \frac{\cos x}{\sin x} \):\[\frac{\csc ^{2} x-\cot ^{2} x}{\sec ^{2} x} = \frac{\frac{1}{\sin^2 x}-\frac{\cos^2 x}{\sin^2 x}}{\frac{1}{\cos^2 x}}\]

Step 2: Simplify the Numerator

Combine the fractions in the numerator over a common denominator:\[\frac{\frac{1-\cos^2 x}{\sin^2 x}}{\frac{1}{\cos^2 x}}\] Since \(1 - \cos^2 x = \sin^2 x\), simplify further:\[\frac{\frac{\sin^2 x}{\sin^2 x}}{\frac{1}{\cos^2 x}} = \frac{1}{\frac{1}{\cos^2 x}}\]

Step 3: Simplify the Overall Expression

Since the numerator simplifies to 1, the expression becomes:\[\frac{1}{\frac{1}{\cos^2 x}} = \cos^2 x\] Thus, the LHS equals the right-hand side (RHS).

Step 4: Verify the Identity

After simplification, we have shown that the LHS of the original identity simplifies to \(\cos^2 x\), which is the same as the RHS. Thus, the identity is verified:\[\frac{\csc ^{2} x-\cot ^{2} x}{\sec ^{2} x} = \cos ^{2} x\]

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc x \), is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function. Therefore, the cosecant of an angle \( x \) can be expressed as the ratio of the hypotenuse to the opposite side in a right triangle. In mathematical form, this is written as:\[ \csc x = \frac{1}{\sin x} \]It's important to note that because \( \csc x \) is the reciprocal of \( \sin x \), it is only defined for angles where \( \sin x eq 0 \). This means \( \csc x \) is undefined for integer multiples of \( \pi \) since at these angles \( \sin x \) equals zero.

• Graph: The graph of \( \csc x \) has vertical asymptotes at every multiple of \( \pi \) because the sine function equals zero there.
• Properties: The cosecant function is periodic with a period of \( 2\pi \).

Understanding \( \csc x \) is crucial in solving trigonometric identities, such as converting between different trigonometric forms to simplify or verify an expression or identity, as demonstrated in the exercise.

Cotangent Function

The cotangent function, symbolized as \( \cot x \), is another trigonometric function, representing the reciprocal of the tangent function. It can be defined as the ratio of the adjacent side to the opposite side in a right triangle. Mathematically, it's represented as:\[ \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \]Like \( \csc x \), \( \cot x \) is undefined when \( \sin x = 0 \), which occurs at integer multiples of \( \pi \).

• Graph: Similar to tangent, the cotangent function graph has vertical asymptotes at multiples of \( \pi \), but shifted by \( \frac{\pi}{2} \).
• Properties: The cotangent function has a period of \( \pi \).

In the context of the exercise, understanding \( \cot^2 x \) helps in simplifying and verifying trigonometric identities. By expressing the cotangent in terms of sine and cosine, it simplifies integration or identity verification, breaking complex expressions into more manageable components.

Secant Function

The secant function, expressed as \( \sec x \), is the reciprocal of the cosine function. It represents the ratio of the hypotenuse to the adjacent side in a right triangle. This function can be mathematically defined as:\[ \sec x = \frac{1}{\cos x} \]\( \sec x \) is undefined where \( \cos x = 0 \), specifically at odd multiples of \( \frac{\pi}{2} \), since the cosine function equals zero at these points.

• Graph: The graph of the secant function shows vertical asymptotes at odd multiples of \( \frac{\pi}{2} \).
• Properties: Secant has a periodicity of \( 2\pi \), similar to the cosine function.

Within the scope of trigonometric identities like the one presented in the exercise, converting \( \sec^2 x \) into cosine form \( \frac{1}{\cos^2 x} \) aids in the simplification process. This method allows you to break down complex trigonometric identities into more straightforward algebraic forms, making solution verification more approachable.