Question

If \( \tan \theta = \dfrac{1}{3} \), then find \( \cos 2\theta \).

• A. \( \dfrac{1}{4} \)
• B. \( \dfrac{1}{10} \)
• C. \( \dfrac{1}{5} \)
• D. \( \dfrac{4}{5} \)

Hint:

When \( \tan \theta \) is given, using the identity for \( \cos 2\theta \) is usually the quickest method.

Answer 1

The Correct Option is D

Step 1: Use the identity for \( \cos 2\theta \).
\[ \cos 2\theta = \frac{1 - \tan^2\theta}{1 + \tan^2\theta} \]

Step 2: Substitute the given value.
\[ \cos 2\theta = \frac{1 - \left(\frac{1}{3}\right)^2}{1 + \left(\frac{1}{3}\right)^2} = \frac{1 - \frac{1}{9}}{1 + \frac{1}{9}} \] \[ = \frac{\frac{8}{9}}{\frac{10}{9}} = \frac{8}{10} \]

Step 3: Simplify.
\[ \cos 2\theta = \frac{4}{5} \]

Step 4: Conclusion.
\[ \boxed{\frac{4}{5}} \]