\cos ^{ 2 }{
\theta } +\dfrac { 1 }{ 1+\cot ^{ 2 }{ \theta } } =\cos ^{ 2
}{ \theta } +\dfrac { \tan ^{ 2 }{ \theta } }{ 1+\tan ^{ 2 }{
\theta } } \quad (\because \cot ^{ 2 }{ \theta } =\dfrac { 1
}{ \tan ^{ 2 }{ \theta } } )\\ =\cos ^{ 2 }{ \theta } +\dfrac
{ \dfrac { \sin ^{ 2 }{ \theta } }{ \cos ^{ 2 }{ \theta
} } }{ \sec ^{ 2 }{ \theta } } \quad (\because \tan {
\theta } =\dfrac { \sin { \theta } }{ \cos { \theta
} } ,\ 1+\tan ^{ 2 }{ \theta } =\sec ^{ 2 }{ \theta } )\\
=\cos ^{ 2 }{ \theta } +\dfrac { \dfrac { \sin ^{ 2 }{ \theta
} }{ \cos ^{ 2 }{ \theta } } }{ \dfrac { 1 }{ \cos ^{ 2 }{
\theta } } } (\because \sec { \theta } =\dfrac { 1 }{
\cos { \theta } } )\\ =\cos ^{ 2 }{ \theta } +\sin ^{ 2 }{
\theta } =1