Animated Solution for Mathematics - Trigonometry: The value of $\tan 9^{\circ }-\tan 27^{\circ }-\tan 63^{\circ }+\tan 81^{\circ }$ is _____.

We are tasked with finding the value of the expression:

When you see a mix of angles like this, your first instinct should be to look for relationships. Notice that the angles $9^{\circ },27^{\circ },63^{\circ }$, and $81^{\circ }$ are related by their sums.

Observe that $9^{\circ }+81^{\circ }=90^{\circ }$ and $27^{\circ }+63^{\circ }=90^{\circ }$. This is not a coincidence; we can group these terms to simplify the expression:

By applying the identity $\tan (90^{\circ }-\theta )=\cot \theta$, we transform the expression into:

We now encounter the expression $\tan \theta +\cot \theta$. This is a classic JEE identity that simplifies as follows:

By multiplying the numerator and denominator by 2, we obtain:

Applying this identity to our grouped expression, we get:

We now substitute the standard values for $\sin 18^{\circ }$ and $\sin 54^{\circ }$:

Substituting these into our expression yields:

Finding the common denominator, the numerator becomes $(\sqrt{5}+1)-(\sqrt{5}-1)=2$, and the denominator becomes $(\sqrt{5}{)}^2-1^2=4$. Thus, the expression simplifies to:

The final value of the expression is 4.