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

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

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

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

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

Solution steps

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

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

Rewrite using trig identities:

cos (\frac{π}{5}) = \frac{5 + 1}{4}

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{2}

Rewrite using trig identities:

sin (\frac{π}{5}) = \frac{2 5 - 5}{4}

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

Simplify

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

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

20 sin (7 2^{\circ})

arcsin (\frac{40}{41})

sec (\frac{3}{2})

arccot (\frac{2}{3})

sin (\frac{3 π}{2}) - cos (\frac{3 π}{2})

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