What is the value of cos(2arctan(9/40)) ?

The value of cos(2arctan(9/40)) is 1519/1681

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cos(2arctan(9/40))

cos (2 arctan (\frac{9}{40}))

\frac{1519}{1681}

Decimal Notation

0.90362 \dots

Solution steps

cos (2 arctan (\frac{9}{40}))

Rewrite using trig identities:

2 cos^{2} (arctan (\frac{9}{40})) - 1

cos (2 arctan (\frac{9}{40}))

Use the Double Angle identity:

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

= 2 cos^{2} (arctan (\frac{9}{40})) - 1

= 2 cos^{2} (arctan (\frac{9}{40})) - 1

Rewrite using trig identities:

cos (arctan (\frac{9}{40})) = \frac{40}{41}

cos (arctan (\frac{9}{40}))

Rewrite using trig identities:

cos (arctan (\frac{9}{40})) = \frac{1 + ( \frac{9}{40} ) ^{2}}{1 + ( \frac{9}{40} ) ^{2}}

Use the following identity:

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

= \frac{1 + ( \frac{9}{40} ) ^{2}}{1 + ( \frac{9}{40} ) ^{2}}

= \frac{1 + ( \frac{9}{40} ) ^{2}}{1 + ( \frac{9}{40} ) ^{2}}

Simplify

= \frac{40}{41}

= 2 (\frac{40}{41})^{2} - 1

Simplify

2 (\frac{40}{41})^{2} - 1 : \frac{1519}{1681}

2 (\frac{40}{41})^{2} - 1

2 (\frac{40}{41})^{2} = \frac{3200}{1681}

2 (\frac{40}{41})^{2}

(\frac{40}{41})^{2} = \frac{4 0 ^{2}}{4 1 ^{2}}

(\frac{40}{41})^{2}

Apply exponent rule:

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

= \frac{4 0 ^{2}}{4 1 ^{2}}

= 2 \cdot \frac{4 0 ^{2}}{4 1 ^{2}}

Multiply fractions:

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

= \frac{4 0 ^{2} \cdot 2}{4 1 ^{2}}

4 0^{2} \cdot 2 = 3200

4 0^{2} \cdot 2

4 0^{2} = 1600

= 1600 \cdot 2

Multiply the numbers:

1600 \cdot 2 = 3200

= 3200

= \frac{3200}{4 1 ^{2}}

4 1^{2} = 1681

= \frac{3200}{1681}

= \frac{3200}{1681} - 1

Convert element to fraction:

1 = \frac{1 \cdot 1681}{1681}

= - \frac{1 \cdot 1681}{1681} + \frac{3200}{1681}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot 1681 + 3200}{1681}

- 1 \cdot 1681 + 3200 = 1519

- 1 \cdot 1681 + 3200

Multiply the numbers:

1 \cdot 1681 = 1681

= - 1681 + 3200

Add/Subtract the numbers:

- 1681 + 3200 = 1519

= 1519

= \frac{1519}{1681}

= \frac{1519}{1681}

\frac{5}{cos ( 4 2 ^{\circ} )}

\frac{sin ^{2} ( 1 )}{1 ^{2}}

sin (1.25)

\frac{sin ( 12 0 ^{\circ} ) cos ( 21 0 ^{\circ} ) tan ( 31 5 ^{\circ} ) cos ( 2 7 ^{\circ} )}{sin ( 6 3 ^{\circ} ) cos ( 54 0 ^{\circ} )}

\frac{π sec ^{2} ( \frac{π}{2} )}{2}

What is the value of cos(2arctan(9/40)) ?
The value of cos(2arctan(9/40)) is 1519/1681