Problem 9 - Algebra and Trigonometry

Question

Verify each identity. \(\cos ^{2} x-\sin ^{2} x=1-2 \sin ^{2} x\)

Step-by-Step Solution

Answer

The given identity \(\cos^2x - \sin^2x = 1 - 2\sin^2x\) is correct.

Step 1: Identify Given Identity

The given identity to be verified is \(\cos^2x - \sin^2x = 1 - 2\sin^2x\). The task is to simplify one side of the equation so that it matches the other side.

Step 2: Substitute Using Trigonometric Identity

Replace \(\cos^2x\) in the left hand side of the equation using the Pythagorean identity \(\cos^2x = 1 - \sin^2x\). The equation becomes \(1 - \sin^2x - \sin^2x\).

Step 3: Simplify Expression

Add like terms to simplify the equation which then results to \(1 - 2\sin^2x\).

Step 4: Compare Both Sides

Now, both sides of the equation match, therefore, verifying the given identity \(\cos^2x - \sin^2x = 1 - 2\sin^2x\).

Key Concepts

Pythagorean identity

The Pythagorean identity is one of the fundamental trigonometric identities. It relates the square of the sine and cosine functions. The identity states that for any angle \(x\), \( \sin^2x + \cos^2x = 1 \). This can be incredibly helpful when simplifying trigonometric expressions, as it allows us to express one trigonometric function in terms of another. In the original exercise, the Pythagorean identity is used to replace \( \cos^2x \) with \( 1 - \sin^2x \).

• It helps in simplifying expressions.
• It serves as a basis for many other trigonometric identities.

This identity always holds true for any real number \(x\), making it a reliable tool in trigonometry problems. Remembering this identity can be a great asset when working on more complex trigonometric proofs and simplifications.

cosine and sine relationship

Understanding the relationship between cosine and sine is fundamental for solving trigonometric equations. These functions are closely related through the Pythagorean identity, which shows that they are essentially phases of each other. In the exercise, we use this relationship to express \( \cos^2x \) in terms of \( \sin^2x \). By doing so, we can unify the expression and simplify it further.
Some key points to remember are:

• If you know either \( \sin x \) or \( \cos x \), you can find the other using the Pythagorean identity.
• Remembering that \( \sin(90^\circ - x) = \cos x \) and \( \cos(90^\circ - x) = \sin x \) can be particularly helpful.

This relationship allows for the seamless transition between sine and cosine, aiding in solving various trigonometric problems.

algebraic manipulation

Algebraic manipulation is a crucial skill in trigonometry, enabling us to simplify and solve equations. In this exercise, we perform algebraic manipulations to verify the identity by matching both sides of the equation through substitution and addition of like terms.
Steps include:

• Substituting \( \cos^2x \) with \( 1 - \sin^2x \) using the Pythagorean identity.
• Combining like terms to simplify the expression to \( 1 - 2\sin^2x \).

By understanding and practicing algebraic manipulation, you can easily rearrange and simplify expressions. This is not only applicable in verifying identities but also in solving various math problems. Mastery of algebraic manipulation enhances your problem-solving skills, particularly where expressions need to be streamlined into simpler forms.