Problem 48 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The identity is verified; both sides equal \( 2\cos^2 x - 1 \).

Step 1: Recall the Pythagorean Identity

The Pythagorean identity states that \( \cos^2 x + \sin^2 x = 1 \). This is fundamental in verifying trigonometric identities.

Step 2: Rewrite the Left Side of the Equation

We start with the left side of the equation: \( \cos^2 x - \sin^2 x \). Use the Pythagorean identity to express \( \sin^2 x \) as \( 1 - \cos^2 x \). Substitute this into the equation to get \( \cos^2 x - (1 - \cos^2 x) \).

Step 3: Simplify the Expression

Simplify \( \cos^2 x - 1 + \cos^2 x \) by combining like terms. This results in \( 2 \cos^2 x - 1 \).

Step 4: Compare the Simplified Expression with the Right Side

Now, compare the simplified expression \( 2 \cos^2 x - 1 \) with the right side of the original equation \( 2 \cos^2 x - 1 \). They are identical, thus verifying the identity.

Key Concepts

Pythagorean identity

The Pythagorean identity is one of the fundamental identities in trigonometry. It is expressed as \(\cos^2 x + \sin^2 x = 1\). This identity is derived from the Pythagorean theorem and is valid for any angle \(x\). It's named after the Greek mathematician Pythagoras because of its connection to the Pythagorean Theorem. This identity allows you to express one trigonometric function in terms of another.
For instance:

• You can write \(\sin^2 x = 1 - \cos^2 x\).
• Similarly, \(\cos^2 x = 1 - \sin^2 x\).

This ability to replace one function with another using this identity is very useful when verifying or simplifying trigonometric expressions. It serves as the groundwork for many proofs and is essential when solving problems involving trigonometric identities.

cosine function

The cosine function, \(\cos x\), is a crucial component of trigonometry. It represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle for an angle \(x\). Cosine is periodic with a period of \(2\pi\), meaning its values repeat every \(2\pi\) radians. The values of the cosine function can range from -1 to 1.
Understanding \(\cos x\) can be visualized on the unit circle:

• In the unit circle, the \(x\)-coordinate of a point gives the value of \(\cos x\).
• The function starts from 1 at \(x = 0\).
• It decreases to 0 at \(x = \pi/2\), then to -1 at \(x = \pi\).
• It continues this symmetrical pattern.

This knowledge is essential when manipulating and verifying trigonometric identities, especially those involving squared expressions like \(\cos^2 x\).

sine function

The sine function, \(\sin x\), is another primary function in trigonometry. It describes the ratio of the opposite side to the hypotenuse in a right-angled triangle. Like cosine, sine is periodic with a period of \(2\pi\). The function oscillates between -1 and 1 across this interval.
Visualize the sine function on the unit circle:

• The \(y\)-coordinate of a point on the circle gives the value of \(\sin x\).
• \(\sin x\) starts from 0 at \(x = 0\).
• It reaches 1 at \(x = \pi/2\), and returns to 0 at \(x = \pi\).
• This pattern mirrors that of cosine but is phase-shifted.

The sine function is integral to trigonometric identities and is often used alongside cosine, as observed in the Pythagorean identity. This function's symmetry and periodic nature make it indispensable for solving trigonometric equations.