# Use ubtraction Formula to write the expression as a trigonometric function of one number.(cos)(13pi)/15cos(-pi/5)-(sin)(13pi)/15sin(-pi/5)

Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value.
$\left(\mathrm{cos}\right)\frac{13\pi }{15}\mathrm{cos}\left(-\frac{\pi }{5}\right)-\left(\mathrm{sin}\right)\frac{13\pi }{15}\mathrm{sin}\left(-\frac{\pi }{5}\right)$
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Bertha Stark
Our expression is
$\left(\mathrm{cos}\right)\frac{13\pi }{15}\mathrm{cos}\left(-\frac{\pi }{5}\right)-\left(\mathrm{sin}\right)\frac{13\pi }{15}\mathrm{sin}\left(-\frac{\pi }{5}\right)$
This kind of expression appears on the addition of angles formula for cosine
$\mathrm{cos}\left(x+y\right)=\mathrm{cos}x\mathrm{cos}y-\mathrm{sin}x\mathrm{sin}y$
with $x=\frac{13\pi }{15},y=-\frac{\pi }{5}$
So we have
$\left(\mathrm{cos}\right)\frac{13\pi }{15}\mathrm{cos}\left(-\frac{\pi }{5}\right)-\left(\mathrm{sin}\right)\frac{13\pi }{15}\mathrm{sin}\left(-\frac{\pi }{5}\right)=\mathrm{cos}\left(\frac{13\pi }{15}+\left(-\frac{\pi }{5}\right)\right)$
$=\mathrm{cos}\left(\frac{13\pi }{15}-\frac{3\pi }{15}\right)$
$=\left(\mathrm{cos}\right)\frac{10\pi }{15}$
$=\left(\mathrm{cos}\right)\frac{2\pi }{3}$
$=-\mathrm{cos}\left(\frac{\pi }{3}\right)\left[\mathrm{cos}\left(\pi -x\right)=-\mathrm{cos}x\right]$
$=-\frac{1}{2}$