# Does tan 75 = cot 75? if so why?

## I was doing a problem in my Trigonometry book and it asked me to "Find an angle (theta) that makes the following statement true." the problem was tan (theta) =cot(45 + 2{theta}) the answer I got confused me it was tan 75 = cot 75

Mar 13, 2017

Given

$\tan \theta = \cot \left(45 + 2 \theta\right)$

$\implies \cot \left(90 - \theta\right) = \cot \left(45 + 2 \theta\right)$

$\implies \left(90 - \theta\right) = \left(45 + 2 \theta\right)$

$\implies 2 \theta + \theta = 90 - 45 = 45$

$\implies 3 \theta = 45$

$\implies \theta = \frac{45}{3} = 15$

Inserting this value in the given condition we get

$\tan 15 = \cot \left(45 + 2 \times 15\right) = \cot 75$

Mar 13, 2017

$\tan {75}^{\circ} = 3.732$
$\cot {75}^{\circ} = 0.269$