What is the value of cos(pi/5)cos(pi/3)-sin(pi/5)sin(pi/3) ?

The value of cos(pi/5)cos(pi/3)-sin(pi/5)sin(pi/3) is (sqrt(5)+1-sqrt(6)sqrt(5-\sqrt{5)})/8

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cos(pi/5)cos(pi/3)-sin(pi/5)sin(pi/3)

cos (\frac{π}{5}) cos (\frac{π}{3}) - sin (\frac{π}{5}) sin (\frac{π}{3})

\frac{5 + 1 - 6 5 - 5}{8}

Solution steps

cos (\frac{π}{5}) cos (\frac{π}{3}) - sin (\frac{π}{5}) sin (\frac{π}{3})

Rewrite using trig identities:

cos (\frac{π}{5}) = \frac{5 + 1}{4}

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

Rewrite using trig identities:

sin (\frac{π}{5}) = \frac{2 5 - 5}{4}

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{2}

= \frac{5 + 1}{4} \cdot \frac{1}{2} - \frac{2 5 - 5}{4} \cdot \frac{3}{2}

Simplify

\frac{5 + 1}{4} \cdot \frac{1}{2} - \frac{2 5 - 5}{4} \cdot \frac{3}{2} : \frac{5 + 1 - 6 5 - 5}{8}

= \frac{5 + 1 - 6 5 - 5}{8}

cos (π - 1)

arctan (\frac{90}{60})

18 sec^{2} (3 π)

sech (2)

\frac{1}{3} cos (0)

What is the value of cos(pi/5)cos(pi/3)-sin(pi/5)sin(pi/3) ?
The value of cos(pi/5)cos(pi/3)-sin(pi/5)sin(pi/3) is (sqrt(5)+1-sqrt(6)sqrt(5-\sqrt{5)})/8