What is the value of cos(2arctan(15/8)) ?

The value of cos(2arctan(15/8)) is -161/289

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cos(2arctan(15/8))

cos (2 arctan (\frac{15}{8}))

- \frac{161}{289}

Decimal Notation

- 0.55709 \dots

Solution steps

cos (2 arctan (\frac{15}{8}))

Rewrite using trig identities:

1 - 2 sin^{2} (arctan (\frac{15}{8}))

cos (2 arctan (\frac{15}{8}))

Use the Double Angle identity:

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

= 1 - 2 sin^{2} (arctan (\frac{15}{8}))

= 1 - 2 sin^{2} (arctan (\frac{15}{8}))

Rewrite using trig identities:

sin (arctan (\frac{15}{8})) = \frac{15}{17}

sin (arctan (\frac{15}{8}))

Rewrite using trig identities:

sin (arctan (\frac{15}{8})) = \frac{( \frac{15}{8} ) 1 + ( \frac{15}{8} ) ^{2}}{1 + ( \frac{15}{8} ) ^{2}}

Use the following identity:

sin (arctan (x)) = \frac{x 1 + x ^{2}}{1 + x ^{2}}

= \frac{( \frac{15}{8} ) 1 + ( \frac{15}{8} ) ^{2}}{1 + ( \frac{15}{8} ) ^{2}}

= \frac{\frac{15}{8} 1 + ( \frac{15}{8} ) ^{2}}{1 + ( \frac{15}{8} ) ^{2}}

Simplify

= \frac{15}{17}

= 1 - 2 (\frac{15}{17})^{2}

Simplify

1 - 2 (\frac{15}{17})^{2} : - \frac{161}{289}

1 - 2 (\frac{15}{17})^{2}

2 (\frac{15}{17})^{2} = \frac{450}{289}

2 (\frac{15}{17})^{2}

(\frac{15}{17})^{2} = \frac{1 5 ^{2}}{1 7 ^{2}}

(\frac{15}{17})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 5 ^{2}}{1 7 ^{2}}

= 2 \cdot \frac{1 5 ^{2}}{1 7 ^{2}}

Multiply fractions:

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

= \frac{1 5 ^{2} \cdot 2}{1 7 ^{2}}

1 5^{2} \cdot 2 = 450

1 5^{2} \cdot 2

1 5^{2} = 225

= 225 \cdot 2

Multiply the numbers:

225 \cdot 2 = 450

= 450

= \frac{450}{1 7 ^{2}}

1 7^{2} = 289

= \frac{450}{289}

= 1 - \frac{450}{289}

Convert element to fraction:

1 = \frac{1 \cdot 289}{289}

= \frac{1 \cdot 289}{289} - \frac{450}{289}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 289 - 450}{289}

1 \cdot 289 - 450 = - 161

1 \cdot 289 - 450

Multiply the numbers:

1 \cdot 289 = 289

= 289 - 450

Subtract the numbers:

289 - 450 = - 161

= - 161

= \frac{- 161}{289}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{161}{289}

= - \frac{161}{289}

sin^{2} (\frac{7 π}{8})

\frac{1}{sin ( 7 5 ^{\circ} )}

sin (2) (1)

π \cdot cos (π)

sec (- 57 0^{\circ})

What is the value of cos(2arctan(15/8)) ?
The value of cos(2arctan(15/8)) is -161/289