Animated Solution for Mathematics - Trigonometry: The value of $(\sin 70^{\circ })(\cot 10^{\circ }\cot 70^{\circ }-1)$ is

The beauty of trigonometric symmetry is often hidden in plain sight. We are tasked with simplifying the expression:

At first, the mix of sine and cotangent might feel like a chaotic jumble, but in mathematics, chaos is often just order waiting to be revealed.

When you encounter a trigonometric expression that feels like a mismatch of functions, the most reliable strategy is to translate everything into the universal language of sine and cosine. We know that $\cot \theta =\frac{\cos \theta }{\sin \theta }$.

By applying this to our cotangent terms, we transform $\cot 10^{\circ }$ into $\frac{\cos 10^{\circ }}{\sin 10^{\circ }}$ and $\cot 70^{\circ }$ into $\frac{\cos 70^{\circ }}{\sin 70^{\circ }}$. Our expression now reads:

Now, we focus on the bracketed term. To subtract the $1$, we need a common denominator, which is $\sin 10^{\circ }\sin 70^{\circ }$.

When we combine the terms, the numerator becomes $\cos 10^{\circ }\cos 70^{\circ }-\sin 10^{\circ }\sin 70^{\circ }$. This is the exact structure of the cosine addition formula:

By identifying $A=10^{\circ }$ and $B=70^{\circ }$, we see that our complex numerator is simply $\cos (10^{\circ }+70^{\circ })$, which is $\cos 80^{\circ }$. Our expression has now simplified to:

The beauty of this step is the immediate, satisfying cancellation of the $\sin 70^{\circ }$ terms in the numerator and denominator. We are left with the simplified fraction:

We are at the final stretch. Recall the complementary angle identity: $\cos \theta =\sin (90^{\circ }-\theta )$.

Applying this to our numerator, $\cos 80^{\circ }$ becomes $\sin (90^{\circ }-80^{\circ })$, which is $\sin 10^{\circ }$. The expression is now:

The complexity has vanished, leaving behind a single, elegant integer. The final answer is 1.