Prove that: (cos(theta)/1-tan(theta)) + sin(theta)/1-cot(theta) = sin( theta) + cos( theta)

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The trigonometric identity `(cos theta)/(1-tan theta) + (sin theta)/(1-cot theta) = sin theta + cos theta ` has to be proved.

`cos(theta)/(1-tan(theta)) + sin(theta)/(1-cot(theta))`

= `cos(theta)/(1-(sin theta)/(cos theta)) + sin(theta)/(1-(cos theta)/(sin theta))`

= `(cos^2theta)/(cos theta - sin theta) + (sin^2 theta)/(sin theta - cos theta)`

= `(cos^2theta)/(cos theta - sin theta) - (sin^2 theta)/(cos theta - sin theta)`

= `(cos^2 theta - sin^2 theta)/(cos theta - sin theta)`

= `((cos theta - sin theta)(cos theta + sin theta))/(cos theta - sin theta)`

= `cos theta + sin theta`

This proves `(cos theta)/(1-tan theta) + (sin theta )/(1-cot theta) = sin theta + cos theta`

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