Problem 12 - Precalculus Essentials

Question

Verify each identity. $$\tan \theta+\cot \theta=\sec \theta \csc \theta$$

Step-by-Step Solution

Answer

The given equation \( \tan \theta + \cot \theta = \sec \theta \csc \theta \) is indeed a true trigonometric identity.

Step 1: Write the terms in terms of sine and cosine

We start by rewriting all terms in terms of \( \sin \theta \) and \( \cos \theta \). We know that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). So, our equation becomes: \( \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = \frac{1}{\cos \theta} \cdot \frac{1}{\sin \theta} \).

Step 2: Simplify the left side of the equation

We proceed by simplifying the left side of the equation. We find a common denominator for the two terms on the left side, which is \( \sin \theta \cos \theta \). So, the equation becomes: \( \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta} \).

Step 3: Use the Pythagorean identity

Next, notice that the numerator of the fraction on the left side is the same as the Pythagorean Identity \( \sin^2 \theta + \cos^2 \theta = 1 \). This simplifies the left side of the equation to \( \frac{1}{\sin \theta \cos \theta} \).

Step 4: Check if left side equals right side

Finally, we can observe that the equation has been simplified such that the left side equals the right side, hence verifying the identity.