Animated Solution for Mathematics - Inverse Trigonometric Functions: If ${\sin }^{-1}\left(\frac{x}{5}\right)+{\text{cosec}}^{-1}\left(\frac{5}{4}\right)=\frac{\pi }{2}$, then the values of $x$ is

Welcome, fellow traveler on the JEE journey. Today, we are going to dissect a problem that, at first glance, might seem like a simple exercise in inverse trigonometry. But beneath the surface, it is a masterclass in the philosophy of problem-solving.

We are looking at the equation:

Our mission is to find the value of $x$. Before we touch a single variable, pause and look at the structure. We have two different inverse functions: ${\sin }^{-1}$ and ${\text{cosec}}^{-1}$. In mathematics, as in life, when you are faced with different 'languages' or 'forms', the first step is always to seek uniformity.

The Translation

Think of ${\text{cosec}}^{-1}$ and ${\sin }^{-1}$ as two sides of the same coin. We know that $\text{cosec}(\theta )=\frac{1}{\sin (\theta )}$. This reciprocal relationship is our golden key.

The property ${\text{cosec}}^{-1}(z)={\sin }^{-1}(\frac{1}{z})$ allows us to translate the complex-looking ${\text{cosec}}^{-1}(\frac{5}{4})$ into something much more familiar. When we apply this, the value inside the function flips. The reciprocal of $\frac{5}{4}$ is $\frac{4}{5}$.

Just like that, our equation transforms:

Suddenly, the equation feels lighter. We have symmetry, and in the JEE Advanced paper, symmetry is often the signal that you are on the right path.

The Identity

Now, look at the right side of the equation: $\frac{\pi }{2}$. Whenever you see $\frac{\pi }{2}$ in an inverse trigonometry problem, your mind should immediately jump to the complementary identity:

This identity is the bedrock of the relationship between sine and cosine. To use it, we rearrange our equation by shifting the ${\sin }^{-1}(\frac{4}{5})$ term to the right side:

Now, look at the right-hand side. It is exactly in the form $\frac{\pi }{2}-{\sin }^{-1}(\theta )$, which is equivalent to ${\cos }^{-1}(\theta )$. So, our equation simplifies beautifully to:

The Triangle Visualization

This is where we bring in the geometry. Let us assume ${\cos }^{-1}(\frac{4}{5})=\theta$. By definition, this means $\cos (\theta )=\frac{4}{5}$.

Imagine a right-angled triangle where the angle is $\theta$. Since $\cos (\theta )=\frac{\text{Base}}{\text{Hypotenuse}}$, we label the base as $4$ and the hypotenuse as $5$. We find the perpendicular side using the Pythagoras theorem:

With the perpendicular as $3$, we find $\sin (\theta )=\frac{\text{Perpendicular}}{\text{Hypotenuse}}=\frac{3}{5}$. This implies that $\theta ={\sin }^{-1}(\frac{3}{5})$. We have successfully converted ${\cos }^{-1}(\frac{4}{5})$ into ${\sin }^{-1}(\frac{3}{5})$.

Substitute this back into our equation:

Since the ${\sin }^{-1}$ function is one-to-one within its principal domain, we equate the arguments directly:

Multiplying both sides by $5$, we arrive at the elegant solution:

Look at what we achieved. We started with a seemingly intimidating equation, used the property of reciprocals to find uniformity, leveraged a fundamental identity to simplify the structure, and used the timeless geometry of a right triangle to reach the finish line. This is the beauty of mathematics—every complex problem is just a series of simple, logical steps waiting to be uncovered.