What is the value of cos^2(pi/8)-sin^2(pi/8) ?

The value of cos^2(pi/8)-sin^2(pi/8) is (sqrt(2))/2

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cos^2(pi/8)-sin^2(pi/8)

cos^{2} (\frac{π}{8}) - sin^{2} (\frac{π}{8})

\frac{2}{2}

Decimal Notation

0.70710 \dots

Solution steps

cos^{2} (\frac{π}{8}) - sin^{2} (\frac{π}{8})

Rewrite using trig identities:

cos (\frac{π}{4})

cos^{2} (\frac{π}{8}) - sin^{2} (\frac{π}{8})

Use the Double Angle identity:

cos^{2} (x) - sin^{2} (x) = cos (2 x)

= cos (2 \cdot \frac{π}{8})

Simplify:

2 \cdot \frac{π}{8} = \frac{π}{4}

2 \cdot \frac{π}{8}

Multiply fractions:

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

= \frac{π 2}{8}

Cancel the common factor:

2

= \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}

sec^{2} (- 24 π) \cdot (- 11) π

tan (arccos (\frac{12}{13}))

\frac{1}{cos ( 6 0 ^{\circ} )}

cos (7 2^{\circ}) cos (13 0^{\circ}) + sin (7 2^{\circ}) sin (13 0^{\circ})

cos (7 0^{\circ}) \cdot cos (4 0^{\circ}) + sin (7 0^{\circ}) \cdot sin (4 0^{\circ})

What is the value of cos^2(pi/8)-sin^2(pi/8) ?
The value of cos^2(pi/8)-sin^2(pi/8) is (sqrt(2))/2