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Prove that: (sec theta - cos theta)(co...

Prove that:
`(sec theta - cos theta)(cot theta + tan theta) = tan theta sec theta`.

Text Solution

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LHS =`(sec theta - cos theta )(cot theta + tan theta)`
`= ((1)/(costheta)-cos theta)((costheta)/(sintheta)+(sintheta)/(costheta))`
`...[sec theta =(1)/(costheta), cottheta=(costheta)/(sintheta) and tantheta = (sintheta)/(costheta)]`
`=((1-cos^2theta)/(costheta))((cos^2theta + sintheta)/(sinthetacostheta))`
`sin^2/costhetaxx1/(sinthetacostheta)`
....[`(therefore sin^2theta+cos^2theta = 1)/(therefore sin^2theta= 1-cos^2theta)]`
`=sintheta/costheta xx 1/costheta`
`=tantheta xx sec theta`
...[`therefore sintheta/costheta = tan theta, 1/costheta = sec theta ]`
=RHS
`therefore` LHS = RHS
`therefore(sec theta - cos theta )(cot theta + tan theta) = tan theta sec theta`.
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