Let $ riangle ABC$ be a right-angled triangle with the right angle at $B$. If $ an A = \sqrt{3}$, then find the values of:

1. $\sin A \cos C + \cos A \sin C$
2. $\cos A \cos C - \sin A \sin C$

Question

Let $	riangle ABC$ be a right-angled triangle with the right angle at $B$. If $	an A = \sqrt{3}$, then find the values of:

1. $\sin A \cos C + \cos A \sin C$
2. $\cos A \cos C - \sin A \sin C$

Accepted Answer

**Solution:**

1. Given $	riangle ABC$ is right-angled at $B$, so $\angle B = 90^\circ$.

2. Since $\angle B = 90^\circ$, the other two angles $A$ and $C$ satisfy:
   $$ A + C = 90^\circ $$

3. Given $	an A = \sqrt{3}$.
   - Recall that $	an 60^\circ = \sqrt{3}$.
   - So, $A = 60^\circ$.

4. Since $A + C = 90^\circ$, then:
   $$ C = 90^\circ - A = 90^\circ - 60^\circ = 30^\circ $$

5. Calculate the first expression:
   $$ \sin A \cos C + \cos A \sin C $$
   - This is the sine addition formula:
   $$ \sin(A + C) $$
   - Since $A + C = 90^\circ$,
   $$ \sin 90^\circ = 1 $$

6. Calculate the second expression:
   $$ \cos A \cos C - \sin A \sin C $$
   - This is the cosine addition formula:
   $$ \cos(A + C) $$
   - Since $A + C = 90^\circ$,
   $$ \cos 90^\circ = 0 $$

**Answer:** $1, 0$

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