Problem 51 - Algebra and Trigonometry

Question

Verify the identity. $$ 2 \cos ^{2} x-1=1-2 \sin ^{2} x $$

Step-by-Step Solution

Answer

Both sides simplify to \(1 - 2\sin^{2}x\), verifying the identity.

Step 1: Recall a Trigonometric Identity

Remember that this identity involves the Pythagorean identity: \[ \ \cos^{2}x + \sin^{2}x = 1 \] We will use this identity to help verify the given equation.

Step 2: Manipulate the Left Side

Start with the left side of the identity given in the problem: \[ \ 2\cos^{2}x - 1 \]Let's use the Pythagorean identity to express \(\cos^{2}x\) in terms of \(\sin^{2}x\). We know \[ \ \cos^{2}x = 1 - \sin^{2}x \]Thus, substitute this into the expression:\[ 2(1 - \sin^{2}x) - 1 \]Distribute the 2:\[ 2 - 2\sin^{2}x - 1 \] Simplify further to:\[ 1 - 2\sin^{2}x \]

Step 3: Compare Both Sides

Upon simplifying the left side, we find that:\[ 1 - 2\sin^{2}x \] is exactly the same as the expression on the right side of the original identity.Thus, both sides of the equation are equal, verifying the identity is correct.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a cornerstone in trigonometry and is derived from the Pythagorean Theorem. Its basic form is \( \cos^{2}x + \sin^{2}x = 1 \).
This identity tells us that for any angle \( x \), the squared values of the sine and cosine functions will sum up to one.

• It's named after the Pythagorean Theorem because the relationship resembles a right triangle's properties.
• This identity is particularly useful when solving and verifying trigonometric identities, as it allows substitution between sine and cosine.

Knowing this identity helps simplify complex trigonometric expressions by converting all terms to a single type of trigonometric function.For instance, when you have a situation where both \( \cos^2 x \) and \( \sin^2 x \) appear, you can often simplify the problem using the Pythagorean Identity.

Cosine and Sine Relationship

Understanding the relationship between cosine and sine is crucial for manipulating trigonometric expressions. Using the Pythagorean Identity, we can express \( \cos^{2}x \) in terms of \( \sin^{2}x \) and vice versa.

• If you know \( \sin^{2}x \), you can find \( \cos^{2}x = 1 - \sin^{2}x \).
• Similarly, \( \sin^{2}x = 1 - \cos^{2}x \).

Example: In the exercise, we started with \( 2\cos^{2}x - 1 \). Using the relationship \( \cos^{2}x = 1 - \sin^{2}x \), we substituted to express it entirely in terms of sine.
This allowed us to simplify the expression and show that both sides of the original equation matched perfectly.
Understanding these conversions is key to successfully verifying trigonometric identities.

Verification of Trigonometric Equations

Verification of trigonometric equations involves proving that both sides of a given equation are equal. It requires a clear understanding of trigonometric identities and relationships among the trigonometric functions.
Here’s how you can approach verifying equations step-by-step:

• Analyze both sides of the equation separately to identify which trigonometric identities can be applied.
• Simplify one or both sides by substituting and manipulating using known identities like the Pythagorean Identity.
• Compare the simplified forms of both sides to check if they match.

In our given exercise, we applied the Pythagorean Identity to replace \( \cos^{2}x \) with \( 1 - \sin^{2}x \) and simplified the expression. When both sides were shown to be the same, the identity was a success!
The ability to verify trigonometric equations is a powerful tool in mathematics, helping confirm that equations hold true for all values within their domain.