Determine all values of $^{'} a^{'}$ for which the equation $c o s^{4} x - (a + 2) c o s^{2} x - (a + 3) = 0$ , possesses solution.Find the solutions.

x = n π \pm s i n^{- 1} \sqrt{a + 3}, a ϵ [- 3, - 2]; n ϵ z

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x = 2 n π \pm c o s^{- 1} \sqrt{a + 3}, a ϵ [- 3, - 2]; n ϵ z

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x = n π \pm c o s^{- 1} (a + 3), a ϵ [- 3, - 2]; n ϵ z

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x = n π \pm c o s^{- 1} \sqrt{a + 3}, a ϵ [- 3, - 2]; n ϵ z

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Solution

The correct option is D $x = n π \pm c o s^{- 1} \sqrt{a + 3}, a ϵ [- 3, - 2]; n ϵ z$
${cos}^{4} x - (a + 2) {cos}^{2} x - (a + 3) = 0$
${cos}^{2} x = \frac{a + 2 \pm \sqrt{{(a + 2)}^{2} + 4 (a + 3)}}{2} = \frac{a + 2 \pm \sqrt{{(a + 4)}^{2}}}{2}$
$= \frac{a + 2 + a + 4}{2}, \frac{a + 2 - a - 4}{2} = a + 3, 0$
${cos}^{2} x = a + 3 \Rightarrow x = n π \pm {cos}^{- 1} \sqrt{a + 3}$
But ${cos}^{- 1} \sqrt{a + 3}$ is defined when $- 1 \leq a + 3 \leq + 1$

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Determine all values of ′a′ for which the equation cos4x−(a+2)cos2x−(a+3)=0, possesses solution.Find the solutions.

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Determine all values of $^{'} a^{'}$ for which the equation $c o s^{4} x - (a + 2) c o s^{2} x - (a + 3) = 0$ , possesses solution.Find the solutions.