If \frac{\tan \\theta}{1-\cot \\theta}+ \frac{\cot \\theta}{1-\tan \\theta}=1+k, then k= _________.

Correct Answer is : cosecθsecθ

SSC CGL 7 06 2019 Shift 1 Paper

Question : 54 of 100

Marks: +1, -0

\ {\tan \θ}/{1-\cot \θ}+\ {\cot \θ}/{1-\tan \θ}=1+k,

then

k=

_________.

Solution:

\ {\tan \θ}/{1-\cot \θ}+\ {\cot \θ}/{1-\tan \θ}

\⇒\ {\ {\sin ^{2} \θ}/{\cos \θ}}/{\sin \θ-\cos \θ}+\ {\ {\cos ^{2} \θ}/{\sin \θ}}/{\cos \θ-\sin \θ}

\⇒\ {1}/{\sin \θ-\cos \θ}\ [\ {\sin ^{2} \θ}/{\cos \θ}-\ {\cos ^{2} \θ}/{\sin \θ}\ ]

\⇒\ {1}{\sin \θ-\cos \θ} \[\ {\sin ^{3} \θ-\cos ^{3} \θ}/{\sin \θ \cos \θ}]

\⇒\ {\sin ^{2} \θ+\cos ^{2} \θ+\sin \θ\cos \θ}/{\sin \θ\cos \θ}

=1+\ {\cosec} \θ\sec \θ

\ {\So}, \ {k}=\ {\cosec} \θ\sec \θ

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