What is the value of cos^2((7pi)/8)-sin^2((7pi)/8) ?

The value of cos^2((7pi)/8)-sin^2((7pi)/8) is (sqrt(2))/2

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cos^2((7pi)/8)-sin^2((7pi)/8)

cos^{2} (\frac{7 π}{8}) - sin^{2} (\frac{7 π}{8})

\frac{2}{2}

Decimal Notation

0.70710 \dots

Solution steps

cos^{2} (\frac{7 π}{8}) - sin^{2} (\frac{7 π}{8})

Rewrite using trig identities:

cos^{2} (\frac{7 π}{8}) = 1 - sin^{2} (\frac{7 π}{8})

cos^{2} (\frac{7 π}{8})

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= 1 - sin^{2} (\frac{7 π}{8})

= 1 - sin^{2} (\frac{7 π}{8}) - sin^{2} (\frac{7 π}{8})

Simplify

= 1 - 2 sin^{2} (\frac{7 π}{8})

Rewrite using trig identities:

sin (\frac{7 π}{8}) = sin (\frac{π}{8})

sin (\frac{7 π}{8})

Use the basic trigonometric identity:

sin (x) = sin (π - x)

= sin (π - \frac{7 π}{8})

Simplify:

π - \frac{7 π}{8} = \frac{π}{8}

π - \frac{7 π}{8}

Convert element to fraction:

π = \frac{π 8}{8}

= \frac{π 8}{8} - \frac{7 π}{8}

Since the denominators are equal, combine the fractions:

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

= \frac{π 8 - 7 π}{8}

Add similar elements:

8 π - 7 π = π

= \frac{π}{8}

= sin (\frac{π}{8})

= 1 - 2 sin^{2} (\frac{π}{8})

Rewrite using trig identities:

cos (\frac{π}{4})

1 - 2 sin^{2} (\frac{π}{8})

Use the Double Angle identity:

1 - 2 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}

sin (\frac{11 π}{20}) cos (\frac{π}{5}) + cos (\frac{11 π}{20}) sin (\frac{π}{5})

cot (arccos (- \frac{5}{6}))

arcsin (0.549)

tan (0.02)

tan (15 0^{\circ}) sin (31 5^{\circ})

What is the value of cos^2((7pi)/8)-sin^2((7pi)/8) ?
The value of cos^2((7pi)/8)-sin^2((7pi)/8) is (sqrt(2))/2