Problem 52 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The given trigonometric identity \( \sin^{4}t - \cos^{4}t = 1 - 2\cos^{2}t \) holds true.

Step 1: Recognize the left side as a difference of squares

Rewrite the left-hand side \(\sin^{4}t - \cos^{4}t\) as a difference of squares. It can be written as \((\sin^{2}t + \cos^{2}t)(\sin^{2}t - \cos^{2}t)\).

Step 2: Use the Pythagorean Identity to simplify the first part

Use the Pythagorean Identity \(\sin^{2}t + \cos^{2}t = 1\) to simplify the first part. This gives the expression \(1(\sin^{2}t - \cos^{2}t)\).

Step 3: Write the expression in another form

Express \(\sin^{2}t - \cos^{2}t\) as \(1 - 2\cos^{2}t\). Therefore, the left-hand side of the equation becomes \(1(1-2\cos^{2}t)\), simplifying to \(1-2\cos^{2}t\). This now matches the right-hand side.

Step 4: Confirm that both sides are equal

Now the equation reads \(1 - 2\cos^{2}t = 1 - 2\cos^{2}t\), verifying that the given identity holds true.