Question

How do you solve this? cosθ-tanθcosθ=0

Answer 1

First, lets note that $tan(\theta)\cdot cos(\theta)=sin(\theta)$. This leads us with the following problem:

$cos(\theta)-sin(\theta)=0$

Lets add sin on both sides, and we get:

$cos(\theta)=sin(\theta)$

Now if we divide with sin on both sides we get:

$\frac{cos(\theta)}{sin(\theta)}=1$

Now we can remember how cot is defined, it is (cos/sin). So we have:

$cot(\theta)=1$

Now take the inverse of cot and we get:
$\theta=cot^{-1}(1)=\pi\cdot n+ \frac{\pi}{4} , \quad n\in \mathbb{Z}$

In general we have $cot^{-1}(1)=\frac{\pi}{4}$, the reason we have to add pi times n, is because it is a function that has multiple answers, see the picture: