# How do I solve this problem?

• Dec 9th 2008, 09:28 AM
lanvin
How do I solve this problem?
Solve each equation for x, where 0 ≤ Θ ≤ 2π

cotx csc
²x = 2cotx

the answer at the back says: π/4, π/2, 3π/4, 5π/4, 3π/2, or 7π/4

How do I solve this?
• Dec 9th 2008, 10:40 AM
Soroban
Hello, lanvin!

Quote:

Solve for $x,\;0 \leq x < 2\pi\!:\quad\cot x \csc^2\!x \:= \:2\cot x$

The answers are: . $\frac{\pi}{4},\;\frac{\pi}{2},\;\frac{3\pi}{4},\;\ frac{5\pi}{4},\;\frac{3\pi}{2},\;\frac{7\pi}{4}$

We have: . $\cot x\csc^2\!x - 2\cot x \;=\;0$

Factor: . $\cot x\bigg[\csc^2\!x - 2\bigg] \;=\;0$

And we have:

. . $\cot x \:=\:0\quad\Rightarrow\quad\boxed{ x \:=\:\frac{\pi}{2},\;\frac{3\pi}{2}}$

. . $\csc^2\!x \:=\:2\quad\Rightarrow\quad \csc x \:=\:\pm\sqrt{2} \quad\Rightarrow\quad\boxed{ x\:=\:\frac{\pi}{4},\;\frac{3\pi}{4},\;\frac{5\pi} {4},\;\frac{7\pi}{4}}$