Problem 18 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

After substituting trigonometric functions and applying the Pythagorean identity, the left-hand side of the equation simplifies to the same expression on the right-hand side, confirming the identity \( \cos t \cot t=\frac{1-\sin ^{2} t}{\sin t} \).

Step 1: Analyze the Identity

Look at the identity \( \cos t \cot t=\frac{1-\sin ^{2} t}{\sin t} \). Start from the left-hand side as it seems to be more straightforward.

Step 2: Rewrite Cotangent

The cotangent function can be written as the reciprocal of the tangent function. So, rewrite \( \cot t \) as \( \frac{1}{\tan t} \). After this step, the left-hand side of the equation will look like this: \( \cos t \cdot \frac{1}{\tan t} \).

Step 3: Express Tangent in terms of Sine and Cosine

The tangent function is equivalent to \(\frac{\sin t}{\cos t}\). Replacing \( \tan t \) with \(\frac{\sin t}{\cos t}\) in the equation, the left-hand side of the equation becomes \( \cos t \cdot \frac{1}{\frac{\sin t}{\cos t}} \).

Step 4: Simplify the Expression

Simplify the expression by multiplying. This will provide \( \frac{\cos t \cdot \cos t}{\sin t} = \frac{\cos^{2} t}{\sin t} \).

Step 5: Convert Cosine-Squared to Sine-Squared

Using the Pythagorean identity, which states that \( \cos^{2} t = 1 - \sin^{2} t \), replace \( \cos^{2} t \) in the equation. This makes the left-hand side of the equation \( \frac{1 - \sin^{2} t}{\sin t} \).

Step 6: Confirm the Identity

Both sides of the equation are confirmed as identical, since \( \frac{1 - \sin^{2} t}{\sin t} = \frac{1 - \sin^{2} t}{\sin t} \). The identity has been verified.