What is the value of cos(2arctan(12/5)) ?

The value of cos(2arctan(12/5)) is -119/169

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cos(2arctan(12/5))

cos (2 arctan (\frac{12}{5}))

- \frac{119}{169}

Decimal

- 0.70414 \dots

Solution steps

cos (2 arctan (\frac{12}{5}))

Rewrite using trig identities:

2 cos^{2} (arctan (\frac{12}{5})) - 1

cos (2 arctan (\frac{12}{5}))

Use the Double Angle identity:

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

= 2 cos^{2} (arctan (\frac{12}{5})) - 1

= 2 cos^{2} (arctan (\frac{12}{5})) - 1

Rewrite using trig identities:

cos (arctan (\frac{12}{5})) = \frac{5}{13}

cos (arctan (\frac{12}{5}))

Rewrite using trig identities:

cos (arctan (\frac{12}{5})) = \frac{1 + ( \frac{12}{5} ) ^{2}}{1 + ( \frac{12}{5} ) ^{2}}

Use the following identity:

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

= \frac{1 + ( \frac{12}{5} ) ^{2}}{1 + ( \frac{12}{5} ) ^{2}}

= \frac{1 + ( \frac{12}{5} ) ^{2}}{1 + ( \frac{12}{5} ) ^{2}}

Simplify

= \frac{5}{13}

= 2 (\frac{5}{13})^{2} - 1

Simplify

2 (\frac{5}{13})^{2} - 1 : - \frac{119}{169}

2 (\frac{5}{13})^{2} - 1

2 (\frac{5}{13})^{2} = \frac{50}{169}

2 (\frac{5}{13})^{2}

(\frac{5}{13})^{2} = \frac{5 ^{2}}{1 3 ^{2}}

(\frac{5}{13})^{2}

Apply exponent rule:

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

= \frac{5 ^{2}}{1 3 ^{2}}

= 2 \cdot \frac{5 ^{2}}{1 3 ^{2}}

Multiply fractions:

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

= \frac{5 ^{2} \cdot 2}{1 3 ^{2}}

5^{2} \cdot 2 = 50

5^{2} \cdot 2

5^{2} = 25

= 25 \cdot 2

Multiply the numbers:

25 \cdot 2 = 50

= 50

= \frac{50}{1 3 ^{2}}

1 3^{2} = 169

= \frac{50}{169}

= \frac{50}{169} - 1

Convert element to fraction:

1 = \frac{1 \cdot 169}{169}

= - \frac{1 \cdot 169}{169} + \frac{50}{169}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot 169 + 50}{169}

- 1 \cdot 169 + 50 = - 119

- 1 \cdot 169 + 50

Multiply the numbers:

1 \cdot 169 = 169

= - 169 + 50

Add/Subtract the numbers:

- 169 + 50 = - 119

= - 119

= \frac{- 119}{169}

Apply the fraction rule:

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

= - \frac{119}{169}

= - \frac{119}{169}

What is the value of cos(2arctan(12/5)) ?
The value of cos(2arctan(12/5)) is -119/169