If 2 tan -1cos-tan -12 cosec- then show that -4- where - is any integer-

We have, 2tan^ 1(cos θ)=tan^ 1(2 cosec θ) ⇒ tant^ 1(2 cos θ/1 cos^2 θ )=tan^ 1(2 cosec θ) [∵ 2 tan^ 1 x=tan^ 1(2x/1 x^2 ) ] ⇒ (2cos θ/sin^2 θ )=(2 ...

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Standard XII

Mathematics

Parametric Differentiation

If 2 tan -1co...

Question

If 2 $t a n^{- 1} (c o s θ) = t a n^{- 1} (2 c o s e c θ), then show that θ = \frac{π}{4},$ where $θ$ is any integer.

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Solution

We have, 2 $t a n^{- 1} (c o s θ) = t a n^{- 1} (2 c o s e c θ)$
$\Rightarrow t a n t^{- 1} (\frac{2 c o s θ}{1 - c o s^{2} θ}) = t a n^{- 1} (2 c o s e c θ) [∵ 2 t a n^{- 1} x = t a n^{- 1} (\frac{2 x}{1 - x^{2}})] \Rightarrow (\frac{2 c o s θ}{s i n^{2} θ}) = (2 c o s e c θ) \Rightarrow (c o t θ .2 c o s e c θ) = (2 c o s e c θ) \Rightarrow c o t θ = 1 \Rightarrow c o t θ = c o t \frac{π}{4} \Rightarrow θ = \frac{π}{4}$

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If 2tan−1(cosθ)=tan−1 (2 cosec θ),then show that θ=π4, where θ is any integer.

We have, 2tan−1(cos θ)=tan−1(2 cosec θ) ⇒ tant−1(2 cos θ1−cos2 θ)=tan−1(2 cosec θ)[∵ 2 tan−1 x=tan−1(2x1−x2)]⇒ (2cos θsin2 θ)=(2 cosec θ)⇒ (cot θ.2 cosec θ)=(2 cosec θ)⇒ cot θ =1⇒ cot θ=cotπ4⇒ θ=π4

If 2 $t a n^{- 1} (c o s θ) = t a n^{- 1} (2 c o s e c θ), then show that θ = \frac{π}{4},$ where $θ$ is any integer.