What is the value of (1-tan^2(15))/(1+tan^2(15)) ?

The value of (1-tan^2(15))/(1+tan^2(15)) is (sqrt(3))/2

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(1-tan^2(15))/(1+tan^2(15))

Solution

\frac{1 - tan ^{2} ( 1 5 ^{\circ} )}{1 + tan ^{2} ( 1 5 ^{\circ} )}

Solution

\frac{3}{2}

Decimal Notation

0.86602 \dots

Solution steps

\frac{1 - tan ^{2} ( 1 5 ^{\circ} )}{1 + tan ^{2} ( 1 5 ^{\circ} )}

Rewrite using trig identities:

tan^{2} (1 5^{\circ}) = \frac{1 - cos ( 3 0 ^{\circ} )}{1 + cos ( 3 0 ^{\circ} )}

tan^{2} (1 5^{\circ})

Use the following identity:

tan^{2} (θ) = \frac{1 - cos ( 2 θ )}{1 + cos ( 2 θ )}

Use the following identity

tan (θ) = \frac{sin ( θ )}{cos ( θ )}

Square both sides

tan^{2} (θ) = \frac{sin ^{2} ( θ )}{cos ^{2} ( θ )}

Rewrite using trig identities:

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

Use the Double Angle identity

cos (2 θ) = 1 - 2 sin^{2} (θ)

Switch sides

2 sin^{2} (θ) - 1 = - cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 - cos (2 θ)

Divide both sides by

2

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

Rewrite using trig identities:

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

Use the Double Angle identity

cos (2 θ) = 2 cos^{2} (θ) - 1

Switch sides

2 cos^{2} (θ) - 1 = cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 + cos (2 θ)

Divide both sides by

2

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

tan^{2} (θ) = \frac{\frac{1 - c o s ( 2 θ )}{2}}{\frac{1 + c o s ( 2 θ )}{2}}

Simplify

tan^{2} (θ) = \frac{1 - cos ( 2 θ )}{1 + cos ( 2 θ )}

= \frac{1 - cos ( 2 \cdot 1 5 ^{\circ} )}{1 + cos ( 2 \cdot 1 5 ^{\circ} )}

Simplify

= \frac{1 - cos ( 3 0 ^{\circ} )}{1 + cos ( 3 0 ^{\circ} )}

= \frac{1 - \frac{1 - c o s ( 3 0 ^{\circ} )}{1 + c o s ( 3 0 ^{\circ} )}}{1 + \frac{1 - c o s ( 3 0 ^{\circ} )}{1 + c o s ( 3 0 ^{\circ} )}}

Simplify

= cos (3 0^{\circ})

Use the following trivial identity:

cos (3 0^{\circ}) = \frac{3}{2}

cos (3 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

= \frac{3}{2}

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Frequently Asked Questions (FAQ)

What is the value of (1-tan^2(15))/(1+tan^2(15)) ?
The value of (1-tan^2(15))/(1+tan^2(15)) is (sqrt(3))/2