Evaluate $\cos \left(2{\cos }^{-1}\frac{5}{13}\right)$.

Step-by-step solution

1. Let $\theta ={\cos }^{-1}\frac{5}{13}$, then $\cos \theta =\frac{5}{13}$.
2. Use the double‑angle identity $\cos 2\theta =2{\cos }^2\theta -1$.
3. Compute ${\cos }^2\theta ={\left(\frac{5}{13}\right)}^2=\frac{25}{169}$.
4. Substitute: $\cos 2\theta =2\cdot \frac{25}{169}-1=\frac{50}{169}-\frac{169}{169}$.
5. Simplify: $\cos 2\theta =-\frac{119}{169}$.
6. Hence $\cos \left(2{\cos }^{-1}\frac{5}{13}\right)=-\frac{119}{169}$.

Concept tested: Double‑angle formula for cosine with inverse cosine

Published 13 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked