Solve the equation for 0<x<360 2csc^2x-cot^4x=-1?

1 Answer
Mar 19, 2018

theta=cos^-1(1/sqrtsqrt2), 2pi-cos^-1(1/sqrtsqrt2)

Explanation:

Given:
2csc^2x-cot^4x=-1
cscx=1/sinx

cotx=cosx/sinx

2(1/sinx)^2-(cosx/sinx)^4=-1

2/sin^2x-cos^4x/sin^4x=-1

Multiplying throughout by sin^4x

2sin^2x-cos^4x=-sin^4x
Transposing

2sin^2x=cos^4x+sin^4x

cos^4x+sin^4x=(cos^2x+sin^2x)^2-2cos^2xsin^2x

2sin^2x=(cos^2x+sin^2x)^2-2cos^2xsin^2x

cos^2x+sin^2x=1

2sin^2x=(1)^2-2cos^2xsin^2x

2sin^2x=1-2cos^2xsin^2x

If
u=cos^2x
1-u=sin^2x

Substituting

2(1-u)=1-2u(1-u)

2-2u=1-2u+2u^2

2u^2-2u+2u+1-2=0

2u^2-1=0

2u^2=1

u^2=1/2

u=+-(1/sqrt2)

u=cos^2x

cos^2x=+-(1/sqrt2)

Considering only positive value for real numbers

cos^2x=1/sqrt2

cosx=+-(1/sqrtsqrt2)

theta=cos^-1(1/sqrtsqrt2), 2pi-cos^-1(1/sqrtsqrt2)