Question

Can anyone help me verify the following identity? cosx/(secx)(sinx)=cscx-sinx

Answer 1

see explanation.

Explanation:

To verify the identity manipulate the left side into the form of the right side.

`"left side "=cosx/(secxsinx)`

`• secx=1/cosxlarrcolor(red)" trig. identity"`

`rArrcosx/(secxsinx)=cosx/(sinx/cosx)=cosx xxcosx/sinx=cos^2x/sinx`

`• cos^2x=1-sin^2xlarrcolor(red)" trig. identity"`

`rArrcos^2x/sinx=(1-sin^2x)/sinx`

`color(white)(xxxxxxxx)=1/sinx-sin^2x/sinx`

`• 1/sinx=cscxlarrcolor(red)"trig. identity"`

`rArr1/sinx-(cancel(sinx)sinx)/cancel(sinx)`

`=cscx-sinx=" right side"rArr" verified"`