What is the value of (1+tan^2(67.5))/(1-tan^2(67.5)) ?

The value of (1+tan^2(67.5))/(1-tan^2(67.5)) is -sqrt(2)

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(1+tan^2(67.5))/(1-tan^2(67.5))

\frac{1 + tan ^{2} ( 67. 5 ^{\circ} )}{1 - tan ^{2} ( 67. 5 ^{\circ} )}

- 2

Solution steps

\frac{1 + tan ^{2} ( 67. 5 ^{\circ} )}{1 - tan ^{2} ( 67. 5 ^{\circ} )}

Rewrite using trig identities:

tan^{2} (67. 5^{\circ}) = \frac{1 - cos ( 13 5 ^{\circ} )}{1 + cos ( 13 5 ^{\circ} )}

= \frac{1 + \frac{1 - c o s ( 13 5 ^{\circ} )}{1 + c o s ( 13 5 ^{\circ} )}}{1 - \frac{1 - c o s ( 13 5 ^{\circ} )}{1 + c o s ( 13 5 ^{\circ} )}}

Simplify

= \frac{1}{cos ( 13 5 ^{\circ} )}

Use the following trivial identity:

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

= \frac{1}{- \frac{2}{2}}

Simplify

\frac{1}{- \frac{2}{2}} : - 2

= - 2

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What is the value of (1+tan^2(67.5))/(1-tan^2(67.5)) ?
The value of (1+tan^2(67.5))/(1-tan^2(67.5)) is -sqrt(2)