Sure! Let's find the exact value for [tex]\( \cos \left(2 \sin^{-1} \frac{7}{10}\right) \)[/tex] step by step.
First, recognize that [tex]\( \sin^{-1} \frac{7}{10} \)[/tex] represents an angle [tex]\(\theta\)[/tex] such that [tex]\( \sin(\theta) = \frac{7}{10} \)[/tex].
### Step 1: Determine [tex]\( \sin(\theta) \)[/tex]
Given:
[tex]\[
\sin(\theta) = \frac{7}{10}
\][/tex]
### Step 2: Calculate [tex]\( \cos(\theta) \)[/tex] using the Pythagorean identity
We know from the Pythagorean identity:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
Thus,
[tex]\[
\cos^2(\theta) = 1 - \sin^2(\theta)
\][/tex]
Substitute the value of [tex]\( \sin(\theta) = \frac{7}{10} \)[/tex]:
[tex]\[
\cos^2(\theta) = 1 - \left(\frac{7}{10}\right)^2
\][/tex]
[tex]\[
\cos^2(\theta) = 1 - \frac{49}{100}
\][/tex]
[tex]\[
\cos^2(\theta) = \frac{100}{100} - \frac{49}{100}
\][/tex]
[tex]\[
\cos^2(\theta) = \frac{51}{100}
\][/tex]
[tex]\[
\cos(\theta) = \sqrt{\frac{51}{100}} = \sqrt{\frac{51}}{10}
\][/tex]
To match the given numerical result, we recognize that:
[tex]\[
\cos(\theta) \approx 0.7141
\][/tex]
### Step 3: Use the double-angle formula for cosine
The double-angle formula for cosine is:
[tex]\[
\cos(2\theta) = 2 \cos^2(\theta) - 1
\][/tex]
### Step 4: Substitute the value of [tex]\( \cos(\theta) \)[/tex] into the double-angle formula
[tex]\[
\cos(2\theta) = 2 \left(\sqrt{\frac{51}}{10}\right)^2 - 1
\][/tex]
Simplify:
[tex]\[
\cos(2\theta) = 2 \left(\frac{51}{100}\right) - 1
\][/tex]
[tex]\[
\cos(2\theta) = 2 \times 0.51 - 1
\][/tex]
[tex]\[
\cos(2\theta) = 1.02 - 1
\][/tex]
[tex]\[
\cos(2\theta) \approx 0.02
\][/tex]
So the exact value of [tex]\( \cos \left(2 \sin^{-1} \frac{7}{10}\right) \)[/tex] is:
[tex]\[
\boxed{0.02}
\][/tex]
