Problem 89 - Algebra and Trigonometry

Question

Verify the identity. $$ (\tan x+\cot x)^{4}=\csc ^{4} x \sec ^{4} x $$

Step-by-Step Solution

Answer

Both sides simplify to \( \csc^4 x \sec^4 x \), verifying the identity.

Step 1: Express Tangent and Cotangent in Sine and Cosine

Recall the trigonometric identities for tangent and cotangent: \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \). Substitute these expressions into the left side of the equation: \( (\tan x + \cot x)^4 = \left( \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} \right)^4 \).

Step 2: Combine Fractions

Combine the fractions inside the parentheses: \( \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} \). Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \), this simplifies to \( \frac{1}{\sin x \cos x} \).

Step 3: Simplify Using Reciprocal Identities

Recognize that \( \frac{1}{\sin x \cos x} = \csc x \sec x \), since \( \csc x = \frac{1}{\sin x} \) and \( \sec x = \frac{1}{\cos x} \). Therefore, we can rewrite the expression as \( (\csc x \sec x)^4 \).

Step 4: Expand the Expression

Expanding \( (\csc x \sec x)^4 \) gives \( \csc^4 x \sec^4 x \).

Step 5: Verify Identity

Since both sides of the given identity simplify to \( \csc^4 x \sec^4 x \), the identity \( (\tan x+\cot x)^{4}=\csc ^{4} x \sec ^{4} x \) is verified.

Key Concepts

Tangent and Cotangent

Understanding tangent and cotangent is crucial when working with trigonometric identities. Tangent is the ratio of the opposite side to the adjacent side in a right triangle, while cotangent is the reciprocal.

• Tangent is given by: \( \tan x = \frac{\sin x}{\cos x} \)
• Cotangent is: \( \cot x = \frac{\cos x}{\sin x} \)

In trigonometric terms, tangent can be thought of as the slope of a line connecting the angle with the origin in the unit circle. Cotangent, being the reciprocal, swaps the roles of sine and cosine, giving you a different perspective on the angle's measure.
When verifying identities, these expressions allow us to interchangeably use sine and cosine, facilitating simplification of equations, like in the original exercise where we substituted these values to simplify the identity.

Sine and Cosine

Sine and cosine are fundamental to trigonometry. They represent the y and x coordinates on the unit circle for a given angle.

• Sine is the opposite side over the hypotenuse: \( \sin x = \frac{\text{opposite}}{\text{hypotenuse}} \)
• Cosine is the adjacent side over hypotenuse: \( \cos x = \frac{\text{adjacent}}{\text{hypotenuse}} \)

These functions give us critical insights into the behavior of angles and are pivotal in proving identities. Their foundational identity, the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \), is frequently used in simplifications and problem-solving exercises. In our exercise, it was crucial in reducing the tangent and cotangent combination to a more manageable form, laying the foundation for further simplification.

Reciprocal Identities

Reciprocal identities extend our understanding of trigonometric functions by relating them to each other through inversion. They are particularly useful for simplifying complex trigonometric expressions.

• Cosecant is the reciprocal of sine: \( \csc x = \frac{1}{\sin x} \)
• Secant is the reciprocal of cosine: \( \sec x = \frac{1}{\cos x} \)

Using these identities, you can transform trigonometric expressions into their reciprocal forms, which can often lead directly to simpler reduced forms. For instance, in the original problem, identifying \( \frac{1}{\sin x \cos x} = \csc x \sec x \) allowed us to rewrite the trigonometric identity, confirming the equality by expanding it to \( \csc^4 x \sec^4 x \). These identities are essential tools in verifying and manipulating trigonometric expressions.