Let S- x- -x- 0-2- The sum of all distinct solutions of the equation -3 sec- x-cosec x-2tan- x-co t-x-0 in the set S is equal to

√(3) sec⁡x+cosec x+2 (tan⁡x cot⁡x )=0 ⇒√(3)/2 sin ⁡x+1/2 cos⁡x=cos^2⁡x sin^2⁡x ⇒ cos⁡ ( x π/3 )=cos⁡2x ⇒ x π/3=2nπ± 2x ⇒ x=2nπ/3+π/9 or x= 2nπ π/3 For xϵ S ...

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Standard XIII

Mathematics

Equations of the Form acosθ+bsinθ = c

Let S= xϵ -π,...

Question

Let $S = {x ϵ (- π, π) : x \neq 0, \pm \frac{π}{2}}$ . The sum of all distinct solutions of the equation $\sqrt{3} s e c x + c o s e c x + 2 (t a n x - c o t x) = 0$ in the set S is equal to

- \frac{7 π}{9}

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- \frac{2 π}{9}

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\frac{5 π}{9}

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Solution

The correct option is C $0$
$\sqrt{3} s e c x + c o s e c x + 2 (t a n x - c o t x) = 0 \Rightarrow \frac{\sqrt{3}}{2} s i n x + \frac{1}{2} c o s x = c o s^{2} x - s i n^{2} x \Rightarrow c o s (x - \frac{π}{3}) = c o s 2 x \Rightarrow x - \frac{π}{3} = 2 n π \pm 2 x \Rightarrow x = \frac{2 n π}{3} + \frac{π}{9} o r x = - 2 n π - \frac{π}{3}$
For $x ϵ S, n = 0 \Rightarrow x = \frac{π}{9}, - \frac{π}{3}$
$n = 1 \Rightarrow x = \frac{7 π}{9}; n = - 1 \Rightarrow x = \frac{- 5 π}{9}$
$∴$ Sum of all values of $x = \frac{π}{9} - \frac{π}{3} + \frac{7 π}{3} - \frac{5 π}{9} = 0$

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Let S={xϵ(−π,π):x≠0,±π2}. The sum of all distinct solutions of the equation √3sec x+cosec x+2(tan x−co tx)=0 in the set S is equal to

The correct option is C 0√3secx+cosecx+2(tanx−cotx)=0⇒√32sin x+12cosx=cos2x−sin2x⇒cos(x−π3)=cos2x⇒x−π3=2nπ±2x⇒x=2nπ3+π9 or x=−2nπ−π3 For xϵS,n=0⇒x=π9,−π3 n=1⇒x=7π9;n=−1⇒x=−5π9 ∴ Sum of all values of x=π9−π3+7π3−5π9=0

Let $S = {x ϵ (- π, π) : x \neq 0, \pm \frac{π}{2}}$ . The sum of all distinct solutions of the equation $\sqrt{3} s e c x + c o s e c x + 2 (t a n x - c o t x) = 0$ in the set S is equal to