Evaluate tan inverse[(3sin2x)/(5+3cos2x)]+tan inverse[(1/4)tanx], where -pi/2

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Q. If

x ϵ (- \frac{π}{2}, \frac{π}{2})

, then the value of

t a n^{- 1} (\frac{t a n x}{4}) + t a n^{- 1} (\frac{3 s i n 2 x}{5 + 3 c o s 2 x})

Q. If

x \in (- \frac{π}{2}, \frac{π}{2})

, then the value of

{tan}^{- 1} (\frac{tan x}{4}) + {tan}^{- 1} (\frac{3 sin 2 x}{5 + 3 cos 2 x})

Tan inverse minus x =-tan inverse x

Oct inverse minus x =π-cot inverse x

Sec inverse minus x =π-sec inverse x

Costco inverse minus x =-cosec inverse x

Inverse circular functions,Principal values of ${sin}^{- 1} x, {c o s}^{- 1} x, {tan}^{- 1} x$ .
${tan}^{- 1} x + {tan}^{- 1} y = {tan}^{- 1} \frac{x + y}{1 - x y}$ , $x y < 1$
$π + {tan}^{- 1} \frac{x + y}{1 - x y}$ , $x y > 1$ .
Evaluate the following :
(a) $sin [\frac{π}{3} - {sin}^{- 1} (- \frac{1}{2})]$
(b) $sin [\frac{π}{2} - {sin}^{- 1} (- \frac{\sqrt{3}}{2})]$

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