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Simplify the expression. Use exact values and show all steps. $$8\left[\left(\cos\dfrac{3\pi}4\right)\left(\sin-\dfrac\pi4\right)\right]+\dfrac1{\sin\frac{5\pi}6}-\left(\tan\pi+\cot\dfrac{7\pi}{4}\right)$$

Here is the answer I got: $$\dfrac{\sin\left(\frac{5\pi}6\right)\left(-\cot\left(\frac{7\pi}4\right)+8\cos\left(\frac{3\pi}4\right)\sin\left(-\frac{\pi}4\right)-\tan(\pi)\right)+1}{\sin\left(\frac{5\pi}6\right)}$$

This is supposed to simplify to $7$, but I'm not quite sure how I can do this?

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  • $\begingroup$ You can evaluate each of those trig functions explicitly. Do so, and plug in the numbers. $\endgroup$ – rogerl Oct 6 '14 at 23:30
  • $\begingroup$ Do you know the values of these trig functions based on the arguments? Do you know what $\sin(\frac{5 \pi}{6})$ is? Do you know how to write all of these in terms of sines and cosines? $\endgroup$ – user28375028 Oct 6 '14 at 23:31
  • $\begingroup$ Use the angle addition formulas, the double and half angle formulas, and known trig angles (like 30, 45, and 60 degrees) together to evaluate the exact value of the functions in your equation. $\endgroup$ – Jonny Oct 6 '14 at 23:34
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There are explicit values inside the trig functions, so you can evaluate the trig functions explicitly as well. $$\cos\left(\frac{3\pi}{4}\right) = \frac{-\sqrt{2}}{2} \\ \sin\left(\frac{-\pi}{4}\right) = \frac{-\sqrt{2}}{2} \\ \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} \\ \tan(\pi) = 0 \\ \cot\left(\frac{7\pi}{4}\right) = -1$$ Hence your quantity becomes $$8\left[\frac{-2}{\sqrt{2}}\cdot \frac{-2}{\sqrt{2}}\right]+\frac{1}{\frac{1}{2}}-\left(0+(-1\right))$$ I believe this will simplify to $19$... If you replace the coefficient of $8$ on the brackets with a $2$ you will get $7$

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