Question

Verify that the following equation is an identity. (1/sinx) - (1/cscx) = cscx - sinx

Answer 1

`\mathrm{csc}\,x = (1)/(\sin x)`, so:


`(1)/(\sin x)-\frac{1}{\mathrm{csc}\,x} = (1)/(\sin x) - (1)/((1)/(\sin x)) = (1)/(\sin x) - \sin x = \mathrm{csc}\,x -\sin x \quad \square`

Answer 2

Manipulating the left side of the equation, we obtain:


`(1)/(\sin x)-(1)/(\csc x)=(\csc x-\sin x)/(\sin x\cdot\csc x)`

Using that
`\csc x=(1)/(\sin x)`:


`(\csc x-\sin x)/(\sin x\cdot\csc x)=(\csc x-\sin x)/(\sin x\cdot(1)/(\sin x))=(\csc x-\sin x)/(1)=\csc x-\sin x\\\\\boxed{(1)/(\sin x)-(1)/(\csc x)=\csc x-\sin x}~~\blacksquare`