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i need to prove: cosec θ-cotθ-1 / 1-cosecθ-cotθ =1-cosθ/sinθ

i need to prove:
cosecθ-cotθ-1 / 1-cosecθ-cotθ
=1-cosθ/sinθ

Grade:10

1 Answers

BALAJI ANDALAMALA
askIITians Faculty 78 Points
5 years ago
cosec\theta-cot\theta-1 = \frac{1}{sin\theta}-\frac{cos\theta}{sin \theta}-\frac{sin\theta}{sin \theta} = \frac{1-cos\theta-sin\theta}{sin\theta}

1-cosec\theta-cot\theta = \frac{sin\theta}{sin\theta}-\frac{1}{sin \theta}-\frac{cos\theta}{sin \theta} = \frac{sin\theta-1-cos\theta}{sin\theta}
L.H.S = \frac{cosec\theta-cot\theta-1}{1-cosec\theta-cot\theta} = \frac{1-cos\theta-sin\theta}{sin\theta-1-cos\theta}

=\frac{2sin^2\frac{\theta}{2}-2sin\frac{\theta}{2}cos\frac{\theta}{2}}{2sin\frac{\theta}{2}cos\frac{\theta}{2}-2cos^2\frac{\theta}{2}}
=\frac{2sin\frac{\theta}{2}(sin\frac{\theta}{2}-cos\frac{\theta}{2})}{2cos\frac{\theta}{2}(sin\frac{\theta}{2}-cos\frac{\theta}{2})} = tan\frac{\theta}{2}

R.H.S = \frac{1-cos\theta}{sin\theta} = \frac{2sin^2\frac{\theta}{2}}{2sin\frac{\theta}{2}cos\frac{\theta}{2}} = tan\frac{\theta}{2}

L.H.S = R.H.S
Hence proved.

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