Q.Using the fact that $\sin (A+B) = \sin A \cos B + \cos A \sin B$ and the differentiation, obtain the sum formula for cosines.

The key idea is to differentiate the sine addition formula to derive the cosine addition formula.

Step 1: Start with the given identity:

Step 2: Differentiate both sides with respect to $A$, treating $B$ as constant. The left side becomes $\cos(A+B) \cdot 1$ by the chain rule. The right side differentiates term-by-term: …

The sine addition formula is given:

We want the cosine addition formula. The key insight is that differentiation links sine and cosine: the derivative of sine is cosine, and the derivative of cosine is negative sine. If we differentiate both sides of the sine formula with respect to $A$ (treating $B$ as a constant), the left side becomes $\cos(A+B)$ by the chain rule, and the right side becomes something we can simplify using the same sine formula again.

Let's work through it step by step.

1. Differentiate both sides with respect to $A$.

   On the left: $\frac{d}{dA} \sin(A+B) = \cos(A+B) \cdot \frac{d}{dA}(A+B) = \cos(A+B) \cdot 1 = \cos(A+B)$.

   On the right: $\frac{d}{dA} [\sin A \cos B + \cos A \sin B] = \cos A \cos B + (-\sin A) \sin B = \cos A \cos B - \sin A \sin B$.

2. Equate the two results.

   Since the derivatives are equal, we have:

That's it — the cosine addition formula emerges directly. …

Method: Deriving One Trig Identity by Differentiating Another

When asked to obtain a new trigonometric identity "using the fact that" another one holds, differentiate the given identity with respect to one of its variables (treating the other as constant) — the chain rule on the left side will reproduce the target identity's structure.

Steps

Step 1: Write down the given identity as an equation between two expressions

Step 2: Choose which variable to differentiate with respect to, and treat the other as a constant

Step 3: Differentiate both sides — apply the chain rule on the left, and the product/sum rule on the right

Step 4: Read off the resulting identity directly from the differentiated equation …

Common Mistakes

Mistake 1: Forgetting the chain-rule factor of $1$ when differentiating $\sin(A+B)$ with respect to $A$.

Why it's wrong: technically $\frac{d}{dA}\sin(A+B)=\cos(A+B)\cdot\frac{d}{dA}(A+B)=\cos(A+B)\cdot1$ — the factor of $1$ is easy to skip mentally since it doesn't change the numeric result, but skipping the reasoning behind it can lead to a wrong factor in a similar problem where the inner derivative isn't $1$. Correct approach: write the chain-rule factor explicitly even when it equals $1$, as a matter of method discipline.

Mistake 2: Losing the minus sign from $\frac{d}{dA}\cos A=-\sin A$ when differentiating the $\cos A\sin B$ term. …