Animated Solution for Mathematics - Trigonometry: $(1+\cos \pi /8)(1+\cos 3\pi /8)(1+\cos 5\pi /8)(1+\cos 7\pi /8)$ is equal to

Unlocking the Trigonometric Product

Welcome, future engineers! Today, we are going to dismantle a problem that, at first glance, looks like a terrifying wall of trigonometric terms. We are asked to evaluate the product:

If you try to calculate these values individually, you will find yourself drowning in nested square roots and half-angle formulas. But in the world of JEE Advanced, we do not fight the problem; we outsmart it. We use symmetry.

Visualizing the Unit Circle

Imagine the unit circle. We have four angles: $\frac{\pi }{8}$, $\frac{3\pi }{8}$, $\frac{5\pi }{8}$, and $\frac{7\pi }{8}$. Look closely at their relationship.

The angles $\frac{\pi }{8}$ and $\frac{7\pi }{8}$ are supplementary—they add up to $\pi$. Similarly, $\frac{3\pi }{8}$ and $\frac{5\pi }{8}$ also add up to $\pi$. This is not a coincidence; it is a geometric invitation.

Recall the fundamental identity $\cos (\pi -\theta )=-\cos \theta$. This tells us that the cosine values of these supplementary angles are mirror images across the y-axis. Specifically:

The Algebraic Collapse

Now, let us substitute these identities back into our original expression. The term $(1+\cos \frac{7\pi }{8})$ transforms into $(1-\cos \frac{\pi }{8})$, and $(1+\cos \frac{5\pi }{8})$ becomes $(1-\cos \frac{3\pi }{8})$.

Our expression now reads:

Do you see the magic? We have created conjugate pairs! By rearranging the terms, we get:

Using the difference of squares identity, $(a+b)(a-b)=a^2-b^2$, this collapses beautifully into:

The Trigonometric Transformation

We are almost there. The expression $1-{\cos }^2\theta$ is the heartbeat of trigonometry: ${\sin }^2\theta$. Our product is now simply:

To solve the inside, we need the product-to-sum formula: $2\sin A\sin B=\cos (A-B)-\cos (A+B)$. We multiply and divide by $2$ to use this identity:

The Final Victory

Let us simplify the angles. The difference is $\frac{2\pi }{8}=\frac{\pi }{4}$, and the sum is $\frac{4\pi }{8}=\frac{\pi }{2}$. We know these values by heart: $\cos \frac{\pi }{4}=\frac{1}{\sqrt{2}}$ and $\cos \frac{\pi }{2}=0$.

Substituting these in, the expression inside the bracket becomes:

Finally, we square this result:

We started with a complex product and, through the elegance of symmetry and identities, arrived at a clean, simple fraction. This is the power of mathematical thinking. Keep practicing, and you will see these patterns everywhere!