Prove that cosec -cosec -1-cosec -cosec -1 - 2 sec2 - -2 MARKS-

Concept: 1 Mark Application: 1 Mark We have =cosec θ/(cosec θ 1)+cosec θ/(cosec θ+1) =cosec θ(cosec θ+1)+cosec θ(cosec θ 1)/(cosec^2 θ 1) =2 cosec^2 θ/(1+co ...

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Standard X

Mathematics

Trigonometric Identity- 3

Prove that co...

Question

Prove that

$\frac{c o s e c θ}{(c o s e c θ - 1)} + \frac{c o s e c θ}{(c o s e c θ + 1)} = 2 s e c^{2} θ$ [2 MARKS]

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Solution

Concept: 1 Mark
Application: 1 Mark

We have

$= \frac{c o s e c θ}{(c o s e c θ - 1)} + \frac{c o s e c θ}{(c o s e c θ + 1)}$

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

$= \frac{2 c o s e c^{2} θ}{(1 + c o t^{2} θ - 1)}$ $[∵ c o s e c^{2} θ = 1 + c o t^{2} θ]$

$= \frac{2 c o s e c^{2} θ}{c o t^{2} θ}$

$= 2 c o s e c^{2} θ t a n^{2} θ$

$= 2 \times \frac{1}{s i n^{2} θ} \times \frac{s i n^{2} θ}{c o s^{2} θ}$

$= \frac{2}{c o s^{2} θ}$

$= 2 s e c^{2} θ$

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Similar questions

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Prove that cosec θ(cosec θ−1)+cosec θ(cosec θ+1)=2 sec2 θ [2 MARKS]

Concept: 1 Mark Application: 1 Mark We have =cosec θ(cosec θ−1)+cosec θ(cosec θ+1) =cosec θ(cosec θ+1)+cosec θ(cosec θ−1)(cosec2 θ−1) =2 cosec2 θ(1+cot2 θ−1) [∵cosec2 θ=1+cot2 θ] =2 cosec2 θcot2 θ =2 cosec2 θ tan2 θ =2×1sin2 θ×sin2 θcos2 θ =2cos2 θ =2 sec2 θ

Prove that

$\frac{c o s e c θ}{(c o s e c θ - 1)} + \frac{c o s e c θ}{(c o s e c θ + 1)} = 2 s e c^{2} θ$ [2 MARKS]