Problem 88 - Algebra and Trigonometry

Question

Verify the identity: $$\sin ^{2} x \tan ^{2} x+\cos ^{2} x \tan ^{2} x=\sec ^{2} x-1$$ (Section 6.1, \text { Example } 3)

Step-by-Step Solution

Answer

The left side of the given equation simplifies down to \(\tan^2x\), so the given identity is verified to be true: \(\tan^2x = \sec^2x - 1\).

Step 1: Simplification

Start by simplifying the given expression. Factor out \(\tan^2x\) from the left side of the equation. It gives us: \(\tan^2x (\sin^2x + \cos^2x)\)

Step 2: Use Pythagorean identity

Now recall the Pythagorean identity \(\sin^2x + \cos^2x = 1\), which can be substituted in place of \(\sin^2x + \cos^2x\). This gives us: \(\tan^2x * 1 = \tan^2x\)

Step 3: Rewrite in terms of Secant

Now, remember the relationship between secant and tangent: \(\sec^2x = 1 + \tan^2x\). Arrange this equation for \(\tan^2x\). This gives us \(\tan^2x = \sec^2x - 1\)

Key Concepts

Pythagorean identity

The Pythagorean identity is one of the most fundamental identities in trigonometry. It states that the sum of the square of sine and the square of cosine is always equal to one: \( \sin^2 x + \cos^2 x = 1 \). This is derived from the Pythagorean theorem and applies to all angles.

• Think of a right triangle in the unit circle, where the radius is 1.
• The sine of an angle is the y-coordinate, and the cosine is the x-coordinate.
• Thus, the identity reflects that both the x and y values (cosine and sine, respectively) will create a relationship where their squares add up to 1.

By using this identity, you can solve many trigonometric problems. It simplifies expressions and helps in verifying other identities. In our problem, it was used to replace \( \sin^2 x + \cos^2 x \) with 1 to simplify the expression.

tangent function

The tangent function, or \( \tan x \), represents the ratio of the sine and cosine functions: \( \tan x = \frac{\sin x}{\cos x} \). It is an essential trigonometric function and has its special properties.

• Tangent is undefined wherever the cosine function is zero, leading to its characteristic vertical asymptotes.
• Tangent function is periodic with a period of \( \pi \) radians, meaning it repeats every 180 degrees.
• It's useful for representing angles of inclination or slopes in various mathematical contexts.

In our exercise, \( \tan^2 x \) was factored out because it's a common term on both sides of the equation. Understanding the tangent's properties and its relation to sine and cosine can aid in manipulating and verifying trigonometric expressions.

secant function

The secant function, noted as \( \sec x \), is the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). It's another vital trigonometric function and closely related to the tangent.

• Just like cosine, the secant function is undefined at certain angles where cosine equals zero. This creates vertical asymptotes at these points.
• It plays a key role in simplifying trigonometric expressions and identities.
• The identity \( \sec^2 x = 1 + \tan^2 x \) directly links secant and tangent, making it useful in conversions and derivations.

In our problem, recognizing that \( \sec^2 x = 1 + \tan^2 x \) allowed the transformation of the equation and confirmed the given identity by rearranging it to solve for \( \tan^2 x \). This demonstrates how interrelated trigonometric functions are and highlights strategies for verifying complex identities.