If 2 tan -1cos-tan -12 cosec- then- show that -4-

2 tan^ 1(cos θ) = tan^ 1 (2 cosec θ) ⇒ tan^ 1 [2cos θ/1 cos^2 θ ]=tan^ 1(2cosec θ) ⇒ tan^ 1(2 cos θ/sin^2 θ )=tan^ 1(2/sin θ ) ⇒2cosθ/sin^2 θ=2/sinθ ⇒ cot θ =1 ...

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Standard X

Mathematics

Trigonometric Ratios

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 $θ = \frac{π}{4}$ .

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Solution

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

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If 2tan−1(cosθ)=tan−1(2cosecθ) then, show that θ=π4.

2tan−1(cosθ)=tan−1(2cosecθ) ⇒tan−1[2cosθ1−cos2θ]=tan−1(2cosecθ) ⇒tan−1(2cosθsin2θ)=tan−1(2sinθ) ⇒2cosθsin2θ=2sinθ ⇒cotθ=1 ⇒tanθ=1 ⇒tanθ=tan(π4) ⇒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 $θ = \frac{π}{4}$ .