Prove that cos(A+B) = cosA·cosB - sinA·sinB using vector method

Question

Using the dot product of two unit vectors, prove that:

\cos(A + B) = \cos A \cos B - \sin A \sin B

(NCERT Class 12, Miscellaneous Exercise)

Solution — Step by Step

Take two unit vectors in the xy-plane:

$\vec{u} = \cos A\,\hat{i} + \sin A\,\hat{j}$ — makes angle $A$ with the x-axis.

$\vec{v} = \cos B\,\hat{i} - \sin B\,\hat{j}$ — makes angle $-B$ with the x-axis.

The angle between $\vec{u}$ and $\vec{v}$ is $A - (-B) = A + B$ .

\vec{u} \cdot \vec{v} = |\vec{u}||\vec{v}|\cos(A + B)

Since both are unit vectors, $|\vec{u}| = |\vec{v}| = 1$ :

\vec{u} \cdot \vec{v} = \cos(A + B)

\vec{u} \cdot \vec{v} = (\cos A)(\cos B) + (\sin A)(-\sin B)

= \cos A \cos B - \sin A \sin B

From Steps 2 and 3:

\boxed{\cos(A + B) = \cos A \cos B - \sin A \sin B}

Hence proved.

Why This Works

The dot product has two equivalent definitions: the geometric one ( $|\vec{u}||\vec{v}|\cos\theta$ ) and the algebraic one (component-wise multiplication and addition). By equating these two forms for cleverly chosen unit vectors, we extract a trigonometric identity.

The trick is in choosing $\vec{v}$ at angle $-B$ (not $+B$ ). This makes the angle between the two vectors equal to $A + B$ . If we had chosen $\vec{v}$ at angle $+B$ , we’d prove the $\cos(A - B)$ formula instead.

Alternative Method — Using rotation matrices

A rotation by angle $A + B$ is equivalent to first rotating by $B$ then by $A$ . The composition of two rotation matrices gives:

R(A+B) = R(A) \cdot R(B)

Expanding the $(1,1)$ entry of both sides yields $\cos(A+B) = \cos A \cos B - \sin A \sin B$ . This approach is more advanced but shows why the identity is fundamentally about rotations.

This vector proof is a favourite in CBSE boards — it appeared in 2019, 2021, and 2023. The proof is short (4 steps), but examiners look for you to clearly state why the angle between $\vec{u}$ and $\vec{v}$ is $A + B$ . Write one sentence explaining the angle choice — it’s worth a mark.

Common Mistake

Students sometimes take both vectors at positive angles: $\vec{u}$ at angle $A$ and $\vec{v}$ at angle $B$ . Then the angle between them is $A - B$ , and you end up proving $\cos(A - B)$ instead of $\cos(A + B)$ . The fix: make one vector at angle $-B$ so the total separation is $A + B$ . Alternatively, prove $\cos(A - B)$ first and then replace $B$ with $-B$ to get $\cos(A + B)$ .

Question

Solution — Step by Step

Why This Works

Alternative Method — Using rotation matrices

Common Mistake

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