Question

If tan x= cot (45 degree+ 2x), what is the value of x

Answer 1

Answer:

$x=\frac{1}{3}(n\pi+\frac{\pi}{4})$

where 'n' is an integer .

Step-by-step explanation:

1)  We have,

$tan(x)=cot(45^{\circ}+2x)\\ \\=>tan(x)=cot(\frac{\pi}{4}+2x)\\ \\=>tan(x)=tan(\frac{\pi}{2}-(\frac{\pi}{4}+2x))\\ \\=>tan(x)=tan(\frac{\pi}{4}-2x)\\ \\=>x = n\pi+(\frac{\pi}{4}-2x)\\ \\=>x+2x=n\pi+\frac{\pi}{4}\\ \\=>3x=n\pi+\frac{\pi}{4}\\ \\=>x=\frac{1}{3}(n\pi+\frac{\pi}{4})$

where 'n' is an integer.

2) However,

If 'x' is an acute angle , then n = 0,1

$x=\frac{1}{3}(n\pi+\frac{\pi}{4})\\ \\=>x=\frac{1}{3}(0+\frac{\pi}{4})=\frac{\pi}{12}=15^{\circ}\\ \\ x=\frac{1}{3}(1*\pi+\frac{\pi}{4})=\frac{5\pi}{12}=75^{\circ}$

Two acute angles are possible , 15 degree and 75 degree.