91Ó°ÊÓ

Trigonometric identities are fundamental tools in mathematics that express relationships between different trigonometric functions such as sine, cosine, and tangent. These identities can simplify complex expressions and solve equations involving trigonometric functions. For example, the identity \( \tan(\frac{\pi}{2} - \beta) = \cot(\beta) \) allows us to convert a tangent function into its cotangent, helping to rewrite expressions.
In solving the given problem, understanding the reciprocal nature of certain trigonometric functions like cosecant \( \operatorname{cosec} (x) \) as \( 1/\sin(x) \) is key. These identities act as bridges that simplify one trigonometric form into another, aiding in the solution process. Trigonometric identities make it possible to transform and manipulate equations for easier resolution.

The property of inverse trigonometric functions is crucial for solving trigonometric equations. These functions, such as \( \tan^{-1}(x) \), are used to find the angle corresponding to a given trigonometric value. The inverse tangent function, for instance, gives an angle whose tangent is \( x \). Such properties help us reverse the process of trigonometric evaluation.
A vital property is \( \tan^{-1}(\tan(x)) = x \), valid under certain conditions, such as \( -\frac{\pi}{2} < x < \frac{\pi}{2} \). This allows us to directly equate the angle with its inverse tangent value, simplifying the left-hand side of trigonometric equations like in the problem. Understanding these properties allows us to methodically reduce and solve otherwise complex expressions.

Algebraic manipulation involves rearranging and simplifying equations to isolate the desired variable, often involving operations like addition, subtraction, multiplication, division, and substitution. By employing algebraic techniques, we can solve the equation step by step once the trigonometric identities and inverse properties have been applied.
In the exercise, after applying the inverse function property, we reformulate the equation as \( 2 \cos x = \frac{\pi}{2} - x \). Solving this requires isolating \( x \) using algebraic methods to balance both sides. Techniques may include first simplifying the expression or substituting known values to streamline the process. This process is integral in arriving at a clear and straightforward solution.