What is the value of cos(pi/(16))cos((3pi)/(16))-sin(pi/(16))sin((3pi)/(16)) ?

The value of cos(pi/(16))cos((3pi)/(16))-sin(pi/(16))sin((3pi)/(16)) is (sqrt(2))/2

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cos(pi/(16))cos((3pi)/(16))-sin(pi/(16))sin((3pi)/(16))

cos (\frac{π}{16}) cos (\frac{3 π}{16}) - sin (\frac{π}{16}) sin (\frac{3 π}{16})

\frac{2}{2}

Decimal Notation

0.70710 \dots

Solution steps

cos (\frac{π}{16}) cos (\frac{3 π}{16}) - sin (\frac{π}{16}) sin (\frac{3 π}{16})

Rewrite using trig identities:

cos (\frac{π}{4})

cos (\frac{π}{16}) cos (\frac{3 π}{16}) - sin (\frac{π}{16}) sin (\frac{3 π}{16})

Use the Angle Sum identity:

cos (s) cos (t) - sin (s) sin (t) = cos (s + t)

= cos (\frac{π}{16} + \frac{3 π}{16})

Simplify:

\frac{π}{16} + \frac{3 π}{16} = \frac{π}{4}

\frac{π}{16} + \frac{3 π}{16}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π + 3 π}{16}

Add similar elements:

π + 3 π = 4 π

= \frac{4 π}{16}

Cancel the common factor:

4

= \frac{π}{4}

= cos (\frac{π}{4})

= cos (\frac{π}{4})

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= \frac{2}{2}

arctan (\frac{6}{4})

sec^{2} (4 5^{\circ})

sin (1 6^{\circ})

sin (1 7^{\circ})

sin (arctan (3))

What is the value of cos(pi/(16))cos((3pi)/(16))-sin(pi/(16))sin((3pi)/(16)) ?
The value of cos(pi/(16))cos((3pi)/(16))-sin(pi/(16))sin((3pi)/(16)) is (sqrt(2))/2