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

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Answer 1 · 2018-03-19T08:20:28

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

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

$cos^2x=1/sqrt2$

$cosx=+-(1/sqrtsqrt2)$

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