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Question
\[\frac{1 - \sin \theta}{\cos \theta}\] is equal to
Options
0
1
sin θ + cos θ
sin θ − cos θ
Solution
The given expression is ` sin θ/(1-cot θ)+ cos θ/(1-tan θ)`
Simplifying the given expression, we have
`sin θ/(1-cot θ)+ cos θ/(1-tan θ)`
= `sinθ/(1-cosθ/sinθ)+cos θ/(1-sinθ/cos θ)`
=` sin θ/((sinθ-cos θ)/sin θ)+cos θ/((cos θ-sin θ)/cos θ)`
= `sin^2θ/(sin θ-cos θ)+cos^2θ/(cos θ-sin θ)`
= `sin^2θ/(sin θ-cos θ)+cos ^2θ/(-1(sin θ-cos θ))`
= `sin ^2θ/(sin θ-cos θ)-cos ^2 θ/(sin θ-cos θ)`
= `(sin^2θ-cos^2θ)/(sin θ-cos θ)`
=` ((sinθ+cos θ)(sinθ-cos θ))/(sin θ-cos θ)`
=` sin θ+cos θ`
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