Sec u tan u cos u1 sin u 1 sec thetatan theta tan theta

Sec u + tan u = cos u/1 - sin u 1 - sec theta/tan theta + tan theta/1 - sec theta = -2 csc theta csc theta + cot theta/tan theta + sin theta = csc theta cot theta

Solution

12) LHS = sec u + tan u = (1/cos u) + (sin u/cos u) = (1 + sin u)/cos u = (1 – sin^2 u)/(1 – sin u)*cos u = cos u/(1-sin u) = RHS

13) LHS = [(1-sec )/tan ] + [tan /(1-sec )] = [(1-sec )^2 + tan^2 ]/(tan *(1-sec ))

          = (1 + sec^2 – 2*sec + tan^2 )/ (tan *(1-sec ))

             = 2sec (sec - 1)/(tan *(1-sec )) = -2*sec /tan = -2*cos /(sin *cos )

          = -2csc = RHS

14) LHS = (csc + cot )/(tan + sin ) = [(1/sin ) + (cos /sin )]/[(sin /cos ) + sin ]

              = [(1+cos )/sin]/[sin*(1 + (1/cos))]

              = (1+cos )*cos /(sin^2 )(1+cos )

              = (cos /sin )*(1/sin ) = cot *csc = RHS