Problem 50 - Precalculus Essentials

Question

Verify each identity. $$\frac{\cos ^{2} t+4 \cos t+4}{\cos t+2}=\frac{2 \sec t+1}{\sec t}$$

Step-by-Step Solution

Answer

Yes. The original trigonometric identity is correct. After simplifying both sides of the equation, we see that they are equal to each other, thereby verifying the identity.

Step 1: Simplify the left side of the equation

First, observe that the expression in the numerator on the left side is a perfect square trinomial: \((\cos t + 2)^2\). This simplifies the left side to \(\frac{(\cos t + 2)^2}{\cos t + 2}\) which simplifies further to \(\cos t + 2\).

Step 2: Convert the right side form in terms of cosine

Convert the right side from secant (\(\sec t\)) form to cosine (\(\cos t\)) form, as we did the simplification in step 1 for the left-hand side in terms of cosine. The right side, \(\frac{2 \sec t+1}{\sec t}\), after rewriting \(\sec t\) as \(\frac{1}{\cos t}\) becomes \(\frac{2 + \cos t}{1}\) which simplifies further to \(2 + \cos t\)

Step 3: Final comparison

From step 1 and step 2, both left side \(\cos t + 2\) and right side \(2 + \cos t\) are equal. So we can conclude that the original trigonometric identity is correct.