Write the value of cosec2-1-cos-1-sin-

Cosec^2 θ (1+cosθ)(1 sinθ)= (1+cos θ)(1 sin θ)/sin^2 θ =(1+cos θ)/sinθ×(1 sinθ)/sinθ = 2cos^2θ/2/2sinθ/2cosθ/2×(cos θ/2 + sinθ/2)^2/2sinθ/2cosθ/2 = 1/2 (1 co ...

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Standard X

Mathematics

Trigonometric Identities

Write the val...

Question

Write the value of $c o s e c^{2} θ (1 + c o s θ) (1 - s i n θ)$ .

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Solution

$c o s e c^{2} θ (1 + c o s θ) (1 - s i n θ)$ = $\frac{(1 + c o s θ) (1 - s i n θ)}{s i n^{2} θ} = \frac{(1 + c o s θ)}{s i n θ} \times \frac{(1 - s i n θ)}{s i n θ} = \frac{2 c o s^{2} \frac{θ}{2}}{2 s i n \frac{θ}{2} c o s \frac{θ}{2}} \times \frac{(c o s \frac{θ}{2} + s i n \frac{θ}{2})^{2}}{2 s i n \frac{θ}{2} c o s \frac{θ}{2}} = \frac{1}{2} (1 - c o t \frac{θ}{2})^{2}$

We have changed the given expression in terms of $c o t \frac{θ}{2}$

So the expression can not be a constant value.

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Write the value of cosec2 θ (1+cos θ)(1−sin θ).

cosec2θ(1+cosθ)(1−sinθ)= (1+cos θ)(1−sin θ)sin2 θ=(1+cos θ)sinθ×(1−sinθ)sinθ=2cos2θ22sinθ2cosθ2×(cosθ2+sinθ2)22sinθ2cosθ2=12(1−cotθ2)2 We have changed the given expression in terms of cotθ2 So the expression can not be a constant value.

Write the value of $c o s e c^{2} θ (1 + c o s θ) (1 - s i n θ)$ .