Question

Verify that this trigonometric equation is an identity?

Cot x sec^4 x = cot x + 2 tan x + tan^3 x

Answer 1

`\cot x\sec^4 x=\cot x+2\tan x+\tan^3x\\\\L=(\cos x)/(\sin x)\cdot(1)/(\cos^4x)=(1)/(\sin x)\cdot(1)/(\cos^3x)=(1)/(\sin x\cos^3x)\\\\R=(\cos x)/(\sin x)+2\cdot(\sin x)/(\cos x)+(\sin^3x)/(\cos^3x)\\\\=(\cos x\cos^3x)/(\sin x\cos^3x)+(2\sin x\cos^2x)/(\cos x\sin x\cos^2x)+(\sin^3x\sin x)/(\cos^3x\sin x)\\\\=(\cos^4x+2\sin^2x\cos^2x+\sin^4x)/(\sin x\cos^3x)\\\\=((\cos^2x)^2+2\sin^2x\cos^2x+(\sin^2x)^2)/(\sin x\cos^3x)`


`=((\cos^2x+\sin^2x)^2)/(\sin x\cos^3x)=(1)/(\sin x\cos^3x)=L\\\\Used:\\\tan(a)=(\sin(a))/(\cos(a))\\\cot(a)=(\cos(a))/(\sin(a))\\\sec(a)=(1)/(\cos(a))\\\sin^2a+\cos^2a=1\\(a+b)^2=a^2+2ab+b^2`