सिद्ध कीजिए कि : `(sec theta + 1 - tan theta)/(tan theta + 1 - sec theta) = (sin theta)/(1 - cos theta) `

Explanation

L.H.S. = `(sec theta + 1 - tan theta)/(tan theta + 1 - sec theta) = ((1)/(cos theta) + 1 - (sin theta)/(1 - cos theta))/((sin theta)/(cos theta ) + 1 - (1)/(cos theta))`
` = (1+ costheta - sin theta)/(sintheta + cos theta - 1) = ((1)/(sin theta) + (cos theta)/(sin theta) - (sin theta)/(sin theta))/((sin theta )/(sin theta )+ (cos theta )/(sin theta ) - (1)/(sin theta)) " "` (अंश और कर को `sin theta` से भाग करने पर )
`("cosec " theta = cot theta - 1)/(1 + cot theta - "cosec " theta ) = ("cosec"^(2) theta - cot ^(2) theta )/(1 + cot theta - "cosec " theta )`
`(("cosec" theta + cot theta )("cosec "theta + cot theta ) ("cosec "theta - cot theta))/(1 + cot theta - "cosec "theta)`
`(("cosec" theta + cot theta )[1" cosec "theta + cot theta ])/(1 + cot theta - "cosec "theta)= "cosec "theta - cot theta`
`(1)/(sin theta)+ (cos theta)/(sin theta) = ((1 + cos theta))/(sin theta) = ((1 + cos theta) (1 - cos theta))/(sin theta (1 - cos theta)`
`(1 - cos^(2) theta)/( sin theta (1 - cos theta) ) = (sin^(2) theta)/(sin theta ( 1 - cos theta)) =(sin theta)/( 1-cos theta ) ` R.H.S.