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Find the exact value of the expression. Write the answer as a single fraction. Do not use a calculator.
\(sin \frac{3 \pi }{2} tan (\frac{-23 \pi }{4}) - cos (\frac{-10 \pi }{3})\)
Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Rationalize all denominators.

Verified Answer :

\(Use:\\sin(x+y)=sinxcosy+sinycosx\\\\tan(-x)=-tanx\\tan(k\pi+x)=tanx\\\\cos(-x)=cosx\\cos(2k\pi+x)=cosx\\cos(x+y)=cosxcosy-sinxsiny\)


\(sin\frac{3\pi}{2}=sin(\pi+\frac{\pi}{2})=sin\pi cos\frac{\pi}{2}+sin\frac{\pi}{2} cos\pi=0\cdot0+1\cdot(-1)=-1\\\\tan\left(-\frac{23\pi}{4}\right)=-tan\frac{23\pi}{4}=-tan\left[6\pi+\left(-\frac{1}{4}\pi\right)\right]=-tan\left(-\frac{1}{4}\pi\right)=tan\frac{\pi}{4}=1\)

\(cos\left(-\frac{10\pi}{3}\right)=cos\frac{10\pi}{3}=cos\left(2\pi+1\frac{1}{3}\pi\right)=cos\left(1\frac{1}{3}\pi\right)=cos\left(\pi+\frac{\pi}{3}\right)\\=cos\pi cos\frac{\pi}{3}-sin\pi sin\frac{\pi}{3}=-1\cdot\frac{1}{2}-0\cdot\frac{\sqrt3}{2}=-\frac{1}{2}\\\\sin\frac{3\pi}{2}tan\left(-\frac{23\pi}{4}\right)-cos\left(-\frac{10\pi}{3}\right)=-1\cdot1-\left(-\frac{1}{2}\right)=-1+\frac{1}{2}=-\frac{1}{2}\)

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