सिद्ध कीजिए कि : ` sin theta (1 + tan theta ) + cos theta (1 + cot theta ) = "cosec" theta + sec theta `

Explanation

L.H.S. = ` sin theta (1 + tan theta) + cos theta (1 + cot theta) `
` = sin theta (1 + tan theta ) + cos theta (1 + (1)/(tan theta ))`
` sin theta 1 + tan theta + cos theta (( tan theta + 1)/(tan theta))`
` = (1 + tan theta) [ sin theta + (cos theta)/(((sin theta)/(cos theta )))] = (1 + tan theta ) (sin theta + (cos^(2) theta )/(sin theta ))`
`= (1 + (sin theta)/(cos theta )) ((sin^(2) theta + cos^(2) theta )/(sin theta) ) = (1 +(sin theta )/(cos theta )) ((1)/(sin theta))`
` (1)/(sin theta) + (sin theta)/(cos theta ) xx (1)/(sin theta) = "cosec " theta + sec theta ` = R.H.S