Answer

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**Hint**: We should know about trigonometry properties to solve such type of problems. Some formulas of trigonometry are needed which are given as-

$

\cos \left( A \right) - \cos \left( B \right) = 2\sin \left( {\dfrac{{A + B}}{2}} \right)\sin \left( {\dfrac{{A - B}}{2}} \right) \\

\sin \theta = 0 \Rightarrow \theta = n\pi \\

$

**:**

__Complete step-by-step answer__The given function is\[\cos \theta = \cos x\].

The general solution for \[\cos \theta = \cos x\]is

\[\cos \theta - \cos x = 0\]

Applying formula$\cos \left( A \right) - \cos \left( B \right) = 2\sin \left( {\dfrac{{A + B}}{2}} \right)\sin \left( {\dfrac{{A - B}}{2}} \right)$ then we get

\[

2\sin \left( {\dfrac{{\theta + x}}{2}} \right)\sin \left( {\dfrac{{\theta - x}}{2}} \right) = 0 \\

\sin \left( {\dfrac{{\theta + x}}{2}} \right) = 0\;{\text{or}}\;\sin \left( {\dfrac{{\theta - x}}{2}} \right) = 0 \\

\]

Further solving we get

\[\left( {\dfrac{{\theta + x}}{2}} \right) = n\pi \;{\text{or}}\;\left( {\dfrac{{\theta - x}}{2}} \right) = n\pi \]

On taking the common solution from both the conditions

\[\theta = 2n\pi \pm x,\;{\text{where}}\;n \in I\]

Thus the answer is true.

**So, the correct answer is “Option A”.**

**Note**: We should be aware of trigonometry formulas. The trigonometric functions are also called circular functions or angle functions. These are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. These functions are used in geometry, navigation, celestial mechanics, solid mechanics and many more. The most widely used trigonometric functions are the ${\text{sine}}$, the ${\text{cosine}}$and the ${\text{tangent}}{\text{.}}$

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