Prove that: \frac{ \sin \theta}{1-\cot \theta}+\frac{\cos \theta}{1-\tan \theta} =\cos \theta+ \sin \theta

ICSE Class X Math 2015 Solved Paper - Question 31

Question : 31 of 52

Marks: +1, -0

Prove that:

\;{ sin \;θ}/{1-\cot θ}+\;{\cos θ}/{1-\tan θ}

=\cos θ+ sin \;θ

Solution:

Given :

\;{ sin \;θ}/{1-\cot θ}+\;{\cos θ}/{1-\tan θ}

=\cos θ+ sin \;θ

L.H.S.

=\;{ sin \;θ}/{1-\;{\cos θ}/{ sin \;θ}}+\;{\cos }/{1-\;{ sin \;θ}/{\cos θ}}

\;=\;{ sin \;^2 θ}/{ sin \;θ-\cos θ}+\;{\cos ^2 θ}/{\cos θ- sin \;θ}

\;=\;{ sin \;^2 θ}/{ sin \;θ-\cos θ}-\;{\cos ^2 θ}/{ sin \;θ-\cos θ}

\;=\;{ sin \;^2 θ-\cos ^2 θ}/{ sin \;θ-\cos θ}

\;=\;{( sin \;θ+\cos θ)( sin \;θ-\cos θ)}/{ sin \;θ-\cos θ}

\;= sin \;θ+\cos θ=\;\text " R.H.S. "\;

L.H.S. = R.H.S.

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