For all \(x \in R\) the number of triplet \((\ell, m, n)\) satisfying equation \(\ell \cos 2 \mathrm{x}+\mathrm{m} \sin ^{2} \mathrm{x}+\mathrm{n}=0\) (a) 2 (b) 4 (c) 6 (d) infinite

Trigonometric functions like sine and cosine are fundamental in mathematics, particularly when dealing with periodic phenomena. The cosine identity we're focusing on here is the double angle identity:
\(\textbf{Cosine Double Angle Identity:}\cos(2x) = 1 - 2\sin^2(x)\).
This identity expresses the cosine of twice an angle in terms of the sine of the angle itself. It's important because it allows us to simplify equations involving trigonometric functions by transforming them into forms where only one type of trigonometric function appears, making the equations more manageable.

Trigonometric identities are equations that are true for all values within their domains. These identities are vital in simplifying and solving trigonometric equations. Some basic examples include:

• Pythagorean Identity: \(\sin^2(x) + \cos^2(x) = 1\)
• Angle Sum and Difference Identities: \(\sin(x \pm y) = \sin(x)\cos(y) \pm \cos(x)\sin(y)\)
• Reciprocal Identities: \(\sin(x) = \frac{1}{\csc(x)}\), \(\cos(x) = \frac{1}{\sec(x)}\)

Using these identities, along with the double angle identities, provides powerful tools for manipulating and simplifying complex trigonometric equations.

Simplifying an equation can often make the difference between an intractable problem and an easily solvable one. The key steps usually involve:

• Consolidating like terms.
• Using algebraic identities to rewrite expressions.
• Factoring, when possible, to reduce equation complexity.

In the context of trigonometric equations, this often means applying trigonometric identities to transform the equation into a standard form where we can more easily identify solutions or make further simplifications.

In mathematics, when we deal with a system of equations, we are looking for values that satisfy all equations in the system simultaneously. In the case of the given exercise, the system is not the set of multiple equations but conditions derived from a single trigonometric equation valid for all real numbers \(x\).
By manipulating the trigonometric equation and using identities, we formed a new system of equations that defined the relationships between the variables ℓ, m, and n. Solving this system gave us the insight needed to determine there are infinite solutions for the triplet (ℓ, m, n), meaning the initial equation has an infinite number of triplets that satisfy it for all \(x\) in \(\mathbb{R}\).