Question

Prove or disprove that csc^4x  − cot^4 x is equal to 2csc^2x  − 1

Answer 1

`^((1))\csc x=(1)/(\sin x)\\\\^((2))\sin^2x+\cos^2x=1\to ^((3))\sin^2x=1-\cos^2x\to ^((4))\cos^2x=1-\sin^2x\\\\^((5))\cot x=(\cos x)/(\sin x)\\\\^((6)) a^2-b^2=(a-b)(a+b)`


`\csc^4x-\cot^4x=2\csc^2x-1\\\\L_s=^((1))(1)/(\sin^4x) -^((5))(\cos^4x)/(\sin^4x)=(1-\cos^4x)/(\sin^4x)=(1^2-(\cos^2x)^2)/(\sin^4x)`


`=^((6))((1-\cos^2x)(1+\cos^2x))/(\sin^4x)=^((3))(\sin^2x(1+\cos^2x))/(\sin^4x)`

`=(1+\cos^2x)/(\sin^2x)=(1)/(\sin^2x)+(\cos^2x)/(\sin^2x)=(1)/(\sin^2x)+^((4))(1-\sin^2x)/(\sin^2x)\\\\=(1)/(\sin^2x)+(1)/(\sin^2x)-(\sin^2x)/(\sin^2x)=(2)/(\sin^2x)-1=R_s`