Animated Solution for Mathematics - Inverse Trigonometric Functions: The numerical value of $\tan \left\{2{\tan }^{-1}\left(\frac{1}{5}\right)-\frac{\pi }{4}\right\}$ is equal to .........

Welcome, future engineer. Today, we are going to dismantle a problem that often intimidates students, not because it is inherently difficult, but because it tests your ability to remain calm and systematic in the face of composite functions.

We are looking at the expression $\tan \{2{\tan }^{-1}(\frac{1}{5})-\frac{\pi }{4}\}$. When you see this, your first instinct might be to panic about the inverse tangent. But I want you to take a deep breath. Mathematics is not about memorizing formulas; it is about recognizing patterns.

The Deconstruction

Let us define our territory. We have an inner angle, $\theta ={\tan }^{-1}(\frac{1}{5})$. By definition, this means $\tan \theta =\frac{1}{5}$.

Imagine this on a coordinate plane. You are standing at the origin, looking at a point $(5,1)$. The angle your line of sight makes with the positive $x$-axis is $\theta$. It is a small, sharp angle.

Now, the expression asks us to evaluate the tangent of $(2\theta -\frac{\pi }{4})$. We have successfully reduced the complexity of the problem. We are no longer dealing with inverse functions; we are dealing with a simple subtraction of angles.

The Bridge

Now, we face the term $2\theta$. How do we find the tangent of a doubled angle when we only know the tangent of the single angle?

This is where the double-angle identity for inverse tangent becomes our best friend. The identity states:

Think of this as a bridge. It allows us to cross from the world of inverse functions into the world of standard algebraic fractions. By substituting $x=\frac{1}{5}$, we get:

Let us pause here. Do not rush the arithmetic. The numerator is $\frac{2}{5}$. The denominator is $1-\frac{1}{25}$, which is $\frac{24}{25}$.

When we divide these, we get:

So, we have discovered that $\tan (2\theta )=\frac{5}{12}$.

The Geometric Reality

Before we proceed to the final step, let us perform a sanity check. We know $\tan (2\theta )=\frac{5}{12}$, which is approximately $0.416$. We also know that $\tan (\frac{\pi }{4})=1$.

Since $0.416<1$, it is geometrically certain that $2\theta <\frac{\pi }{4}$. This confirms that our final result must be negative.

If you ever get a positive answer, you know immediately that you have made a sign error. This is the power of geometric intuition—it acts as a safety net for your algebra.

The Final Calculation

We are now at the finish line. We need to evaluate $\tan (2\theta -\frac{\pi }{4})$. We invoke the subtraction formula:

Substituting $A=2\theta$ and $B=\frac{\pi }{4}$, we get:

Look at the elegance of this expression. The numerator becomes $\frac{5-12}{12}=-\frac{7}{12}$. The denominator becomes $\frac{12+5}{12}=\frac{17}{12}$.

When we divide these, the common denominator of $12$ cancels out beautifully, leaving us with $-\frac{7}{17}$.

You see? The complexity melts away when you follow the logic. You have just navigated a multi-step trigonometric problem with precision. Keep this confidence. Every time you face a complex expression, break it down, find the bridge, and trust the process. You are ready for the next challenge.