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Question
Prove that `tan(pi/4 + theta) tan((3pi)/4 + theta)` = – 1
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Solution
`tan(pi/4 + theta) = (tan pi/4 + tan theta)/(1 - tan pi/4 * tan theta)`
`tan(pi/4 + theta) = (1 + tan theta)/(1 - tan theta)` .....(1)
`tan ((3pi)/4 +theta) = (tan (3pi)/4 + tan theta)/(1 - tan (3pi)/4 * tan theta)`
= `(tan(pi - pi/4) tan theta)/(1 - tan(pi - pi/4) tan theta)`
= `(- tan pi/4 + tan theta)/(1 + tan pi/4 * tan theta)`
`tan((3pi)/4 + theta) = (-1 + tan theta)/(1 + tan theta)` .....(2)
From equation (1) and (2) we have
`tan(pi/4 + theta) * tan((3pi)/4 + theta)`
= `(1 + tan theta)/(1 - tan theta) xx (-(1 - tan theta))/(1 + tan theta)`
= – 1
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