Problem 34 - Algebra 2

Question

Simplify each trigonometric expression. $$ \cos \theta+\sin \theta \tan \theta $$

Step-by-Step Solution

Answer

The simplified form of the trigonometric expression \( \cos \theta + \sin \theta \cdot \tan \theta \) is \( \sec \theta \).

Step 1: Rewrite tan in terms of sine and cosine

Start by replacing \( \tan \theta \) with \( \frac{\sin \theta}{\cos \theta} \) to simplify the equation. The equation now looks like this: \( \cos \theta + \sin \theta \cdot \frac{\sin \theta}{\cos \theta} \)

Step 2: Simplify the Expression

Now multiply \( \sin \theta \) with \( \frac{\sin \theta}{\cos \theta} \) to simplify the equation. The equation will then be \( \cos \theta + \frac{ \sin^2 \theta}{\cos \theta} \)

Step 3: Create a Common Denominator

In order to add these terms, they must have the same denominator. So multiply \( \cos \theta \) by \( \cos \theta \) to have the same denominator. \( \frac {\cos^2 \theta}{\cos \theta} + \frac {\sin^2 \theta}{\cos \theta} \)

Step 4: Combine the fractions

With the common denominator, the fractions can now be combined into one. The equation now becomes \( \frac{\cos^2 \theta + \sin^2 \theta}{\cos \theta} \)

Step 5: Substitute the Pythagorean Identity

In the numerator, substitute the Pythagorean Identity, namely \( \sin^2 \theta + \cos^2 \theta = 1 \). The equation simplifies to \( \frac{1}{\cos \theta} \)

Step 6: Rewrite expression with cosine

Finally, rewrite \( \frac{1}{\cos \theta} \) as \( \sec \theta \). This is the simplest form of the original expression.