solve the equation for an exact solution. sin⁻¹x - tan⁻¹1 = -π/3 choose the correct answer below. oa. {(-√6…

Explanation:

Step1: Evaluate $\tan^{-1}1$

We know that $\tan^{-1}1=\frac{\pi}{4}$. So the equation $\sin^{-1}x - \tan^{-1}1=-\frac{\pi}{3}$ becomes $\sin^{-1}x-\frac{\pi}{4}=-\frac{\pi}{3}$.

Step2: Isolate $\sin^{-1}x$

Add $\frac{\pi}{4}$ to both sides of the equation: $\sin^{-1}x=-\frac{\pi}{3}+\frac{\pi}{4}$.

Step3: Calculate the right - hand side

Find a common denominator. $-\frac{\pi}{3}+\frac{\pi}{4}=\frac{-4\pi + 3\pi}{12}=-\frac{\pi}{12}$. So $\sin^{-1}x=-\frac{\pi}{12}$.

Step4: Solve for $x$

Take the sine of both sides. $x = \sin(-\frac{\pi}{12})$. Since $\sin(-\alpha)=-\sin\alpha$, then $x=-\sin\frac{\pi}{12}$. And $\sin\frac{\pi}{12}=\frac{\sqrt{6}-\sqrt{2}}{4}$, so $x =-\frac{\sqrt{6}-\sqrt{2}}{4}=\frac{-\sqrt{6}+\sqrt{2}}{4}$.

Answer:

A. $\left{\frac{-\sqrt{6}+\sqrt{2}}{4}\right}$