Let's solve the given expressions step-by-step.
### (a) \(\sin \left(\sin ^{-1} \frac{7}{9}\right)\)
The expression \(\sin(\sin^{-1}(x))\) is essentially the sine function composed with its inverse. The inverse sine function, \(\sin^{-1}(x)\), returns an angle \(\theta\) such that \(\sin(\theta) = x\) and \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\).
Given the expression:
[tex]\[
\sin \left(\sin^{-1} \left(\frac{7}{9}\right)\right)
\][/tex]
we are essentially looking for the sine of the angle whose sine is \(\frac{7}{9}\). By the definition of inverse functions:
[tex]\[
\sin \left(\sin^{-1} \left(\frac{7}{9}\right) \right) = \frac{7}{9}
\][/tex]
Thus, the value is:
[tex]\[
\boxed{\frac{7}{9}}
\][/tex]
### (b) \(\cos \left[\cos ^{-1}\left(-\frac{7}{9}\right)\right]\)
Similarly, the expression \(\cos(\cos^{-1}(x))\) is the cosine function composed with its inverse. The inverse cosine function, \(\cos^{-1}(x)\), returns an angle \(\theta\) such that \(\cos(\theta) = x\) and \(0 \leq \theta \leq \pi\).
Given the expression:
[tex]\[
\cos \left(\cos^{-1} \left(-\frac{7}{9}\right)\right)
\][/tex]
we are looking for the cosine of the angle whose cosine is \(-\frac{7}{9}\). By the definition of inverse functions:
[tex]\[
\cos \left(\cos^{-1} \left(-\frac{7}{9}\right)\right) = -\frac{7}{9}
\][/tex]
Thus, the value is:
[tex]\[
\boxed{-\frac{7}{9}}
\][/tex]
### (c) \(\tan \left[\tan ^{-1}(-10)\right]\)
The expression \(\tan(\tan^{-1}(x))\) is the tangent function composed with its inverse. The inverse tangent function, \(\tan^{-1}(x)\), returns an angle \(\theta\) such that \(\tan(\theta) = x\) and \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\).
Given the expression:
[tex]\[
\tan \left(\tan^{-1}(-10)\right)
\][/tex]
we are looking for the tangent of the angle whose tangent is \(-10\). By the definition of inverse functions:
[tex]\[
\tan \left(\tan^{-1}(-10)\right) = -10
\][/tex]
Thus, the value is:
[tex]\[
\boxed{-10}
\][/tex]
### Summary
The exact values of the given expressions are:
(a) \(\boxed{\frac{7}{9}}\)
(b) \(\boxed{-\frac{7}{9}}\)
(c) [tex]\(\boxed{-10}\)[/tex]
