Answer

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**Hint:**For solving the given expression take $\cos x$as common from the expression given in the

question, after taking common equalise and compare the equations to determine the general

solution of $\cos x$.

As, general solution for $\cos x = 0$ is $x = \dfrac{{\left( {2n + 1} \right)\pi }}{2},$ apply this to solve the expression given in the question.

**Complete step by step solution:**As per data given in the question,

We have,

$2{\cos ^2}x + \cos x = 0...(i)$

Here from equation $(i)$

Taking $\cos x$ as common.

We will get,

$\therefore \cos x\left( {2\cos x + 1} \right) = 0$

Hence, from above expression we can conclude that,

Either first part will be equal to zero or second part will be equal to zero.

So,

Either $\cos x = 0$ or $2\cos x + 1 = 0$

If, $\cos x = 0$

If, $2\cos x + 1 = 0$

Then,$\cos x = \dfrac{{ - 1}}{2}$

Hence, general solution for $\cos x = 0$ will be,

$x = \dfrac{{\left( {2n + 1} \right)\pi }}{2},$ where $n$ is integer and general solution for \[\cos x = \dfrac{{ - 1}}{2} = \cos \left( { \pm \dfrac{{2\pi }}{3}} \right)\] is $x = n\pi \pm \dfrac{{2\pi }}{3}$, where $n$ is an integer.

Hence General solution for $2{\cos ^2}x + \cos x = 0$ will be,

$n = \dfrac{{\left( {2n + 1} \right)\pi }}{2}$ or \[x = 2\pi \pm \dfrac{{2\pi }}{3}\]

Where $n$ is an integer.

**Additional Information:**

From the trigonometric triangle,

We have,

Here, AC is adjacent or base of triangle and BC is altitude or Opposite of triangle and AC is hypotenuse of triangle.

So, from here,

We have,

Angle of sine will be \[ = \operatorname{Sin} \theta = \dfrac{{Base}}{{Hypotenuse}} = \dfrac{b}{c}\]

Angle of Cosine will be \[ = \operatorname{Cos} \theta = \dfrac{{Opposite}}{{Hypotenuse}} = \dfrac{a}{c}\]

Angle of tangent will be \[ = \operatorname{Tan} \theta = \dfrac{{Base}}{{Opposite}} = \dfrac{b}{a}\]

Hence, from the above trigonometric triangle we can determine the relationship between all trigonometric variables.

Value of the angle of cosine decreases from 0 to 90, and after 90 it becomes negative.

Value of the angle of secant is inverse of the value of cosine.

Value of the angle of cosecant is the inverse of the value of the angle of Sine.

Value of the angle of Cot is the inverse of the value of angle of tangent.

**Note:**

As here,

From question,

As, $\cos x\left( {2\cos x + 1} \right) = 0$

Hence, from above expression we can conclude that,

Either the first part will be equal to zero or the second part will be equal to zero.

So,

The general equation of the expression given in the question will be an addition of the general equation of both the values.

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