If sin (a + b) = 1 and cos (a – b) = $\frac{1}{2}$, then find a.

The correct answer is 75°.

We are given two trigonometric conditions:

Condition 1: sin(a + b) = 1
Condition 2: cos(a - b) = $\frac{1}{2}$

We need to find the value of a using standard trigonometric values.

Step 1: Solve Condition 1

We know from standard trigonometry that sin(90°) = 1.

So, if sin(a + b) = 1, then:

$a+b=90°$    ...(i)

Step 2: Solve Condition 2

We know from standard trigonometry that cos(60°) = $\frac{1}{2}$.

So, if cos(a - b) = $\frac{1}{2}$, then:

$a-b=60°$    ...(ii)

Step 3: Add equations (i) and (ii)

Now we add both equations together to eliminate b:

$(a+b)+(a-b)=90°+60°$

$2a=150°$

$a=\frac{150}{2}=75°$

Step 4: Verify

Substituting a = 75° back into equation (i):
75° + b = 90° ⇒ b = 15°

Check Condition 1: sin(75° + 15°) = sin(90°) = 1 ✓
Check Condition 2: cos(75° - 15°) = cos(60°) = $\frac{1}{2}$ ✓

Both conditions are satisfied, confirming that a = 75°.