# If sin θ + cos θ = 2–√ cos (90°−θ), find cot θ.

Question:

If $\sin \theta+\cos \theta=\sqrt{2} \cos \left(90^{\circ}-\theta\right)$, find $\cot \theta$.

Solution:

Given: $\sin \theta+\cos \theta=\sqrt{2} \cos \left(90^{\circ}-\theta\right)$

We have to find the value of $\cot \theta$.

Now,

$\sin \theta+\cos \theta=\sqrt{2} \cos \left(90^{\circ}-\theta\right)$

$\Rightarrow \sin \theta+\cos \theta=\sqrt{2} \sin \theta$ (since, $\cos \left(90^{\circ}-\theta\right)=\sin \theta$ )

$\Rightarrow \cos \theta=(\sqrt{2}-1) \sin \theta$

$\Rightarrow \frac{\cos \theta}{\sin \theta}=\sqrt{2}-1$

$\Rightarrow \cot \theta=\sqrt{2}-1$

Hence, $\cot \theta=\sqrt{2}-1$