Animated Solution for Mathematics - Inverse Trigonometric Functions: The value of $\tan (2{\tan }^{-1}(\frac{3}{5})+{\sin }^{-1}(\frac{5}{13}))$ is equal to:

Imagine you are standing at the threshold of a complex trigonometric landscape. You are presented with the expression $\tan (2{\tan }^{-1}(\frac{3}{5})+{\sin }^{-1}(\frac{5}{13}))$.

At first glance, it looks like a chaotic mix of inverse functions. However, our goal is to transform this expression into a language that the outer $\tan$ function understands perfectly.

We want everything inside that bracket to be a ${\tan }^{-1}$ function. Because $\tan ({\tan }^{-1}x)=x$, this is the ultimate simplification.

Let us focus on the first part: $2{\tan }^{-1}(\frac{3}{5})$. We have a coefficient of $2$ that is blocking our path.

We reach into our toolkit and pull out the double-angle identity for inverse tangent:

By substituting $x=\frac{3}{5}$, the expression becomes:

The numerator is $\frac{6}{5}$, and the denominator is $1-\frac{9}{25}=\frac{16}{25}$. When we divide $\frac{6}{5}$ by $\frac{16}{25}$, we are effectively multiplying $\frac{6}{5}$ by $\frac{25}{16}$.

Simplifying this, we are left with ${\tan }^{-1}(\frac{15}{8})$. The first beast is tamed.

Now, we turn to ${\sin }^{-1}(\frac{5}{13})$. This is where we use our geometric intuition.

Let $\theta ={\sin }^{-1}(\frac{5}{13})$, which implies $\sin \theta =\frac{5}{13}$. Picture a right-angled triangle where the perpendicular is $5$ and the hypotenuse is $13$.

Using the Pythagorean theorem, the base is:

Since $\tan \theta =\frac{\text{Perpendicular}}{\text{Base}}$, we find that $\tan \theta =\frac{5}{12}$. Thus, ${\sin }^{-1}(\frac{5}{13})={\tan }^{-1}(\frac{5}{12})$. We have achieved total uniformity.

Our expression is now $\tan ({\tan }^{-1}(\frac{15}{8})+{\tan }^{-1}(\frac{5}{12}))$. We apply the tangent addition formula:

Here, $A={\tan }^{-1}(\frac{15}{8})$ and $B={\tan }^{-1}(\frac{5}{12})$. Substituting these, we get:

Calculating the numerator:

Calculating the denominator:

Finally, we divide the results:

We have arrived at the summit. The complexity has dissolved into the final answer: $\frac{220}{21}$.