Given $\tan\theta=\frac{5}{12}$ and 0 < $\theta$ < $\frac{\pi}{2}$, find $\cos\left(2\theta\right)$.

• A. 0
• B. $-\frac{199}{169}$
• C. $\frac{119}{169}$
• D. $\frac{144}{169}$

Answer: (C)

Verified step by step guidance

1. Start by recalling the identity for tangent: \( \tan\theta = \frac{\text{opposite}}{\text{adjacent}} \). Given \( \tan\theta = \frac{5}{12} \), you can consider a right triangle where the opposite side is 5 and the adjacent side is 12.
2. Use the Pythagorean theorem to find the hypotenuse of the triangle: \( c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \).
3. Now, find \( \cos\theta \) using the definition of cosine: \( \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13} \).
4. Recall the double angle identity for cosine: \( \cos(2\theta) = 2\cos^2\theta - 1 \). Substitute \( \cos\theta = \frac{12}{13} \) into the identity: \( \cos(2\theta) = 2\left(\frac{12}{13}\right)^2 - 1 \).
5. Calculate \( \left(\frac{12}{13}\right)^2 = \frac{144}{169} \) and substitute into the identity: \( \cos(2\theta) = 2 \times \frac{144}{169} - 1 \). Simplify this expression to find \( \cos(2\theta) \).