Is tan(pi/4-θ)=(1-tan(θ))/(1+tan(θ)) ?

The answer to whether tan(pi/4-θ)=(1-tan(θ))/(1+tan(θ)) is True

Popular >

prove tan(pi/4-θ)=(1-tan(θ))/(1+tan(θ))

prove

tan (\frac{π}{4} - θ) = \frac{1 - tan ( θ )}{1 + tan ( θ )}

True

Solution steps

tan (\frac{π}{4} - θ) = \frac{1 - tan ( θ )}{1 + tan ( θ )}

Manipulating left side

tan (\frac{π}{4} - θ)

Rewrite using trig identities

= \frac{cos ( θ ) - sin ( θ )}{cos ( θ ) + sin ( θ )}

Manipulating right side

\frac{1 - tan ( θ )}{1 + tan ( θ )}

Express with sin, cos

= \frac{cos ( θ ) - sin ( θ )}{cos ( θ ) + sin ( θ )}

We showed that the two sides could take the same form

\Rightarrow True

\frac{sin ( θ )}{cot ( θ ) + csc ( θ )} = 1 - cos (θ)

csc^{2} (x) cos^{2} (x) = cot^{2} (x)

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

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

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

Is tan(pi/4-θ)=(1-tan(θ))/(1+tan(θ)) ?
The answer to whether tan(pi/4-θ)=(1-tan(θ))/(1+tan(θ)) is True