Animated Solution for Mathematics - Trigonometry: Let $n$ be an odd integer. If $\sin n\theta ={\sum }_{r=0}^n{b}_r{\sin }^r\theta$, for every value of $\theta$, then

We are presented with the trigonometric identity:

This expression is a functional identity, meaning it holds true for all values of $\theta$. Because it is an identity, we are free to choose specific values for $\theta$ or apply operations like differentiation to isolate the coefficients $b_r$.

To isolate the constant term $b_0$, we examine the expanded form:

Notice that every term on the right side, except for $b_0$, contains at least one factor of $\sin \theta$. By choosing $\theta =0$, we force all terms involving $\sin \theta$ to zero.

Substituting $\theta =0$ into the identity yields:

Thus, we find that $b_0=0$.

To isolate $b_1$, we cannot simply set $\theta =0$, as $b_1\sin (0)$ would also vanish. Instead, we employ the power of calculus by differentiating both sides of the identity with respect to $\theta$.

Applying the chain rule to the left side and the power rule to the right side, we obtain:

Now, we substitute $\theta =0$ into this differentiated identity. Every term on the right side containing a factor of $\sin \theta$ will vanish:

Since $\cos (0)=1$, the equation simplifies to:

Therefore, we conclude that $b_1=n$.

By utilizing the properties of identities, we successfully determined the coefficients without resorting to complex expansions. Through the strategic use of substitution and differentiation, we found:

$b_0=0$ and $b_1=n$.

Remember that in competitive mathematics, the most elegant path is often found by manipulating the structure of the equation rather than brute-forcing the algebra.