If cosec -2- show that -sin-1-cos-2-

We know that cotθ = cosθ/sinθ cot θ + sin θ/1 + cos θ = cosθ/sinθ + sin θ/1 + cos θ = cos θ(1 + cos θ) + sin^2 θ/sin θ(1 + cos θ) = cos θ + cos^2 θ + sin^2 θ/s ...

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Standard X

Mathematics

Relation between Trigonometric Ratios

If cosec θ=2,...

Question

If cosec $θ = 2$ , show that $c o t θ + \frac{s i n θ}{1 + c o s θ}$ = 2.

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Solution

We know that $c o t θ = \frac{c o s θ}{s i n θ}$

$c o t θ + \frac{s i n θ}{1 + c o s θ}$

$= \frac{c o s θ}{s i n θ} + \frac{s i n θ}{1 + c o s θ}$

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

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

$∵ c o s^{2} θ + s i n^{2} θ = 1$

= $\frac{c o s θ + 1}{s i n θ (1 + c o s θ)}$

= $\frac{1}{s i n θ}$

= $c o s e c θ = 2$

Hence proved.

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If cosec θ=2, show that cotθ+sinθ1+cosθ = 2.

We know that cotθ=cosθsinθ cotθ+sinθ1+cosθ =cosθsinθ+sinθ1+cosθ = cosθ(1+cosθ)+sin2θsinθ(1+cosθ) = cosθ+cos2θ+sin2θsinθ(1+cosθ) ∵cos2θ+sin2θ=1 = cosθ+1sinθ(1+cosθ) =1sinθ = cosecθ=2 Hence proved.

If cosec $θ = 2$ , show that $c o t θ + \frac{s i n θ}{1 + c o s θ}$ = 2.