Problem 10 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

After step 1 (rearranging), step 2 (applying known trigonometric identities), and step 3 (simplifying the equation), we have shown that \(\cos ^{2} x-\sin ^{2} x=2 \cos ^{2} x-1\) is indeed a trigonometric identity.

Step 1: Re-arrange the equation

Start by re-arranging the right side of original equation so that it can be easier to compare the two sides. Original equation:\(\cos ^{2} x-\sin ^{2} x=2 \cos ^{2} x-1\)Re-arranged equation:\(\cos ^{2} x-\sin ^{2} x=\cos ^{2} x+ \cos ^{2} x -1\)

Step 2: Apply known trigonometric identities

Next, replace \(\cos ^{2} x + \cos ^{2} x\) on the right side of the equation by using the Pythagorean identity. As \(\sin ^{2} x + \cos ^{2} x = 1\), then \(\cos ^{2} x + \cos ^{2} x = 1 - \sin ^{2} x\). Implementing this into the equation gives:\(\cos ^{2} x - \sin ^{2} x = 1 - \sin ^{2} x - 1\)

Step 3: Simplify the Equation

After implementing step 2, the equation simplifies to: \(\cos ^{2} x - \sin ^{2} x = \cos ^{2} x - \sin ^{2} x\)This confirms that the left side of the equation is the same as the right side. This implies that the original equation has been properly verified

Key Concepts

Pythagorean Identity

Pythagorean Identity is a fundamental concept in trigonometry that relates the squares of sine and cosine of an angle. It's based on the Pythagorean Theorem from geometry, which describes the relationship between the sides of a right triangle. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Similarly, the Pythagorean Identity in trigonometry is expressed as: \[ \sin^2 x + \cos^2 x = 1 \] This identity is incredibly useful because it allows us to express one trigonometric function in terms of the other. For example, if you know the value of \( \sin x \), you can easily find \( \cos x \) using this identity. In the given problem, this identity helps in transforming expressions involving \( \cos^2 x \) and \( \sin^2 x \). By rearranging terms and substituting using the Pythagorean Identity, the equations become simpler and logical, aiding in verifying the trigonometric identity at hand.

Cosine

Cosine is one of the primary trigonometric functions. It represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. If we have an angle \( x \), cosine of \( x \), written as \( \cos x \), is the length of the side adjacent to the angle, divided by the length of the hypotenuse. Cosine plays a crucial role not just in basic trigonometry but also in complex formulas. It appears in various identities and equations, helping to simplify and resolve them. In the problem statement, \( \cos^2 x \) (which means \( \cos x \) squared) is used to express and verify a trigonometric identity. Knowing that \( \cos^2 x \) can be manipulated and combined with \( \sin^2 x \) via identities like the Pythagorean Identity, enables us to solve and confirm the correctness of our equations efficiently.

Sine

Sine is another fundamental trigonometric function used to describe the ratio between the opposite side and the hypotenuse in a right triangle. For a given angle \( x \), the sine of \( x \), denoted as \( \sin x \), is the length of the side opposite the angle divided by the hypotenuse. The sine function, like cosine, is pivotal in solving trigonometric equations and in establishing relationships between angles and sides in triangles. In the context of our problem, \( \sin^2 x \) (which is \( \sin x \) squared) is used to demonstrate and verify the equation. \( \sin^2 x \) is inherently linked with \( \cos^2 x \) as shown in the Pythagorean Identity \( \sin^2 x + \cos^2 x = 1 \). By substituting \( \sin^2 x \) in place of corresponding expressions, we simplify and prove the equations, ensuring that both sides of the verified identity match.