To find the exact value of the expression [tex]\(\tan \left(\sin^{-1} \frac{12}{13} + \cos^{-1}(-1)\right)\)[/tex], follow these steps:
1. Understand the given values:
[tex]\[
\sin^{-1} \frac{12}{13} \quad \text{and} \quad \cos^{-1}(-1)
\][/tex]
2. Evaluate [tex]\(\cos^{-1}(-1)\)[/tex]:
By definition, [tex]\(\cos^{-1}(-1)\)[/tex] is the angle whose cosine is [tex]\(-1\)[/tex]. The angle whose cosine is [tex]\(-1\)[/tex] in the principal range [tex]\([0, \pi]\)[/tex] is [tex]\(\pi\)[/tex]:
[tex]\[
\cos^{-1}(-1) = \pi
\][/tex]
3. Evaluate [tex]\(\sin^{-1} \frac{12}{13}\)[/tex]:
The angle whose sine value is [tex]\(\frac{12}{13}\)[/tex] in the principal range [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex]. Denote this angle as [tex]\(\theta\)[/tex]:
[tex]\[
\sin \theta = \frac{12}{13}
\][/tex]
Using the Pythagorean identity [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex], we can determine [tex]\(\cos \theta\)[/tex]:
[tex]\[
\cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{12}{13}\right)^2 = 1 - \frac{144}{169} = \frac{25}{169}
\][/tex]
[tex]\[
\cos \theta = \sqrt{\frac{25}{169}} = \frac{5}{13}
\][/tex]
Thus, [tex]\(\theta\)[/tex] satisfies:
[tex]\[
\theta = \sin^{-1} \frac{12}{13}
\][/tex]
4. Sum the angles:
Next, we sum the angles that we have:
[tex]\[
\sin^{-1} \frac{12}{13} + \cos^{-1}(-1) = \theta + \pi
\][/tex]
5. Evaluate [tex]\(\tan \left(\theta + \pi \right)\)[/tex]:
The tangent function has the property that [tex]\(\tan(x + \pi) = \tan x\)[/tex]:
[tex]\[
\tan(\theta + \pi) = \tan \theta
\][/tex]
6. Determine [tex]\(\tan \theta\)[/tex]:
Recall that:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{5}
\][/tex]
Thus, the exact value of the expression [tex]\(\tan \left(\sin^{-1} \frac{12}{13} + \cos^{-1}(-1)\right)\)[/tex] is:
[tex]\[
\boxed{\frac{12}{5}}
\][/tex]
