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Question
Prove that `tan (pi/4 + theta) - tan(pi/4 - theta)` = 2 tan 2θ
Solution
`tan (pi/4 + theta) - tan(pi/4 - theta)` = `(tan pi/4 + tan theta)/(1 - tan pi/4 tan theta) - (tan pi/4 - tan theta)/(1 + tan pi/4 tan theta)`
= `(1 + tan theta)/(1 - tan theta) - (1 - tan theta)/(1 + tan theta)`
= `((1 + tan theta)^2 - (1 - tan theta)^2)/((1 - tan theta) (1 + tan theta))`
= `((1 + 2 tan theta + tan^2theta) - (1 - 2 tan theta + tan^2theta))/(1 - tan^2theta)`
= `(1 + 2 tan theta + tan^2theta - 1 + 2tan theta - tan^2theta)/(1 - tan^2theta)`
= `(4 tan theta)/(1 - tan^2theta)`
= `2 * (2 tan theta)/(1 - tan^2theta)`
= 2 tan 2θ
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