Find the exact value of the expression.
-$\cos \left[ 2 \sin ^ { - 1 } \left( - \frac { 5 } { 13 } \right) \right]$
A) $\frac { 119 } { 169 }$
B) $\frac { 10 } { 13 }$
C) $- \frac { 12 } { 13 }$
D) $\frac { 2 \sqrt { 5 } + 10 } { 13 }$

Question

Quizplus · Accepted Answer

The expression can be simplified using the double-angle formula for cosine, $\cos(2\theta) = 1 - 2\sin^2(\theta)$, and the given $\sin^{-1}(-\frac{5}{13})$ implies that $\sin(\theta) = -\frac{5}{13}$. Substituting $\theta$ with $\sin^{-1}(-\frac{5}{13})$, we get $\cos(2\theta) = 1 - 2\left(-\frac{5}{13}\right)^2 = 1 - 2\left(\frac{25}{169}\right) = 1 - \frac{50}{169} = \frac{119}{169}$.

Find the Exact Value of the Expression cos⁡[2sin⁡−1(−513)]\cos \left[ 2 \sin ^ { - 1 } \left( - \frac { 5 } { 13 } \right) \right]cos[2sin−1(−135​)]

Find the Exact Value of the Expression $\cos \left[ 2 \sin ^ { - 1 } \left( - \frac { 5 } { 13 } \right) \right]$