Problem 51 - Precalculus Essentials

Question

Verify each identity. $$\cos ^{4} t-\sin ^{4} t=1-2 \sin ^{2} t$$

Step-by-Step Solution

Answer

Yes, the identity \(\cos^{4}(t)-\sin^{4}(t)=1-2 \sin^{2}(t)\) has been verified successfully. The detailed steps showcase the application of multiple trigonometric identities to solve the given exercise.

Step 1: Recognize the form

The function \(\cos^{4}(t) - \sin^{4}(t)\) appears to be in the form of the difference of squares. Thus, it can be written as \((a^2 - b^2) = (a + b)(a - b)\). Applying this, we have \((\cos^2(t) + \sin^2(t))(\cos^2(t) - \sin^2(t))\).

Step 2: Use Pythagorean identity

Using the Pythagorean identity, \(\sin^2(t) + \cos^2(t)\) is equal to 1. So, the equation becomes \(1(\cos^2(t) - \sin^2(t))\).

Step 3: Convert to single trigonometric function

We can convert \(\cos^2(t) - \sin^2(t)\) into a single trigonometric function by using the identity \(cos(2t) = cos^2(t) - sin^2(t)\). Therefore, the right-hand side becomes \(cos(2t)\).

Step 4: Use power-reduction identity

We can rewrite \(cos(2t)\) using power-reduction identity, \(cos(2t) = 1 - 2sin^2(t)\). Thus, applying this, the right-hand side becomes \(1 - 2sin^2(t)\).

Step 5: Confirm match

Now, the left-hand side \(\cos^{4}(t) - \sin^{4}(t)\) has been manipulated to become \(1 - 2sin^2(t)\), which matches the right-hand side of the original equation. So, the identity has been verified.