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Question
Which of the following options is equal to cot θ+cosec θ−1cot θ−cosec θ+1
Which of the following options is equal to cot θ+cosec θ−1cot θ−cosec θ+1
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Solution
The correct option is D cotθ+cosecθ
=cot θ+cosec θ−1cot θ−cosec θ+1
=cot θ+cosec θ−(cosec2θ−cot2θ)cot θ−cosec θ+1
=(cot θ+cosec θ)−(cosec θ−cot θ)(cosec θ+cot θ)cot θ−cosec θ+1
=(cot θ+cosec θ)(1−cosec θ+cot θ)(cot θ−cosec θ+1)
=cot θ+cosec θ
=cos θsin θ+1sin θ
=1+cos θsin θ
=(1+cos θ)(1−cos θ)sinθ(1−cos θ) ....[Multiplying numerator and denominator by 1−cosθ ]
=1−cos2θsinθ(1−cos θ)
=sin2θsinθ(1−cos θ)
=sin θ1−cos θ
cotθ+cosecθ
=cot θ+cosec θ−1cot θ−cosec θ+1
=cot θ+cosec θ−(cosec2θ−cot2θ)cot θ−cosec θ+1
=(cot θ+cosec θ)−(cosec θ−cot θ)(cosec θ+cot θ)cot θ−cosec θ+1
=(cot θ+cosec θ)(1−cosec θ+cot θ)(cot θ−cosec θ+1)
=cot θ+cosec θ
=cos θsin θ+1sin θ
=1+cos θsin θ
=(1+cos θ)(1−cos θ)sinθ(1−cos θ) ....[Multiplying numerator and denominator by 1−cosθ ]
=1−cos2θsinθ(1−cos θ)
=sin2θsinθ(1−cos θ)
=sin θ1−cos θ
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