Find the value of tan-1-2sin -12 x -1- x 2-cos -11- y 2-1- y 2- x -0- xy -1-

X = tan A,y = tan B ⇒ tan [ 1/2{ sin^ 1 ( 2x/1+x^2 )+ cos^ 1 ( 1 y^2/1+y^2 ) }], = tan [ 1/2{ sin^ 1(sin 2A) + cos^ 1(cos 2B) } ] = tan [ 1/2(2A + 2B) ] = t ...

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Byju's Answer

Standard XII

Mathematics

Trigonometric Ratios of Multiples of an Angle

Find the valu...

Question

Find the value of tan $[\frac{1}{2} {s i n^{- 1} (\frac{2 x}{1 + x^{2}}) + c o s^{- 1} (\frac{1 - y^{2}}{1 + y^{2}})}], | x | < 1, y > 0, x y < 1$ .

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Solution

$x = t a n A, y = t a n B$
$\Rightarrow t a n [\frac{1}{2} {s i n^{- 1} (\frac{2 x}{1 + x^{2}}) + c o s^{- 1} (\frac{1 - y^{2}}{1 + y^{2}})}],$
$= t a n [\frac{1}{2} {s i n^{- 1} (s i n 2 A) + c o s^{- 1} (c o s 2 B)}]$
$= t a n [\frac{1}{2} (2 A + 2 B)]$
$= t a n (A + B)$
$= \frac{t a n A + t a n B}{1 - t a n A t a n B} = \frac{x + y}{1 - x y}$

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Find the value of tan [12{sin−1(2x1+x2)+cos−1(1−y21+y2)}],|x|<1,y>0,xy<1 .

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Find the value of tan $[\frac{1}{2} {s i n^{- 1} (\frac{2 x}{1 + x^{2}}) + c o s^{- 1} (\frac{1 - y^{2}}{1 + y^{2}})}], | x | < 1, y > 0, x y < 1$ .