The general value $[\theta ]$ is obtained from the equation$[Cos2\theta = {\mathop{\rm Sin}\nolimits} \alpha ] $,is
A. $[2\theta = \pi /2 - \alpha ]$
B.$ [\theta = 2n\pi \pm (\pi /2 - \alpha )]$
C. $[\theta = n\pi + {( - 1)^n}\alpha /2]$
D. $[\theta = n\pi \pm (\pi /4 - \alpha /2)]$

Hint: It is a field of mathematics that examines how triangles' side lengths and angles relate to one another. A trigonometric equation is one that has one or more ratios of unknown trigonometric angles. The ratios of sine, cosine, tangent, cotangent, secant, and cosecant angles are used to express it. A trigonometric equation is $[\cos 2x + 5\sin x = 0]$, for instance. solution of the trigonometric equation refers to all possible values that fulfill the given trigonometry.

Formula Used: $[\cos \theta = \cos \alpha \Rightarrow \theta = 2n\pi \pm \alpha ]$

Complete step-by-step solution: The general value is obtained from the equation$[{\mathop{\rm Cos}\nolimits} 2\theta = {\mathop{\rm Sin}\nolimits} \alpha \]$ ,is
$[\cos 2\theta = \cos (\pi /2 - \alpha )]$
$[ \Rightarrow 2\theta = 2n\pi \pm (\pi /2 - \alpha )]$
$[ \Rightarrow \theta = n\pi \pm (\pi /4 - \alpha /2)]$
Hence, the correct option is option D.

A main solution is the unknown angle's smallest numerical value satisfying the equation in the range $[0 \le x \le 2\pi ]$. To define the primary answer, we will use the range $[0 \le x \le 2\pi ]$. Furthermore, there may be two solutions at this time. Both answers are legitimate, but we only choose the one that is quantitatively more advantageous. This aids in defining the trigonometric functions' principal domain so that their inverses can be obtained.

Option ‘D’ is correct

Note: Application of a Model's Properties to Unfamiliar Situations: Mistakes Errors caused by hastily substituting one thing for another. This obviously falls into the third category, in my opinion. The pupil switched the $[10]$ and the $[x]$ since their brain was working so hard.
I find these kinds of errors to be particularly intriguing since I believe many teachers will notice them and comment, "Oy, this student thinks you can just switch out the x with the angle." Or someone else can comment, "Wow, this student doesn't even understand the basics of trigonometry.