91Ó°ÊÓ

One of the tools you'll frequently use in trigonometry is the double angle identities. These are equations that express trigonometric functions of double angles, like \(2x\), in terms of single angles, like \(x\). The cosine double angle identity is especially useful: \[ \cos 2x = \cos^2 x - \sin^2 x \].

There are multiple forms of the cosine double angle identity:

• \( \cos 2x = 2\cos^2 x - 1 \)
• \( \cos 2x = 1 - 2\sin^2 x \)
• \( \cos 2x = \frac{1 - \tan^2 x}{1 + \tan^2 x} \)

These variations can simplify complex trigonometric expressions or help solve trigonometric equations.
In the given problem, we use the third form to verify that the identity \( \frac{1-\tan^2 x}{1+\tan^2 x} = \cos 2x \) is correct.
This transformation shows the flexibility and power of trigonometric identities in solving equations.

The Pythagorean identity is a cornerstone in trigonometry, similar to the famous Pythagorean theorem for triangles. It states that:
\[ \sin^2 x + \cos^2 x = 1 \]
This can be manipulated into other useful forms by dividing by \(\cos^2 x\) or \(\sin^2 x\), resulting in:

• \(1 + \tan^2 x = \sec^2 x \)
• \(1 + \cot^2 x = \csc^2 x \)

The identity \(1 + \tan^2 x = \sec^2 x\) plays a crucial role in the exercise's second step. It bridges the expression of tangent with secant, allowing us to substitute \(\sec^2 x\) for \(1 + \tan^2 x\), simplifying the equation. Recognizing and applying this identity can be the key to simplifying and solving many trigonometric problems.

Trigonometric simplification involves rewriting expressions in a simpler or more convenient form using known identities. In this exercise, the expression \( \sec^2 x - 2\tan^2 x \) needs to be simplified.

To do this, we use the identity \( \sec^2 x = 1 + \tan^2 x \). Substituting this expression allows us to rewrite\

• \( \sec^2 x - 2\tan^2 x = (1 + \tan^2 x) - 2\tan^2 x = 1 - \tan^2 x \)

This simplification shows that the expression can become \( \frac{1 - \tan^2 x}{1 + \tan^2 x} \).
Simplifying trigonometric expressions is about cleverly applying identities to make the math easier to handle. It all boils down to understanding how different trigonometric functions relate to one another. This understanding is crucial for verifying identities and solving equations.