Correct option is: D) None of these

(A) A = \(30^\circ\) and B = \(60^\circ\)

= A + B = \(30^\circ\) + \(60^\circ\) = \(90^\circ\)

Now, sin A + sin B = sin \(30^\circ\) + sin \(60^\circ\)

= \(\frac 12\) + \(\frac {\sqrt3}{2}\)

= \(\frac {1+\sqrt3}{2} >1\)

But sin (A +B) = sin \(90^\circ\) = 1

Hence, sin A + sin B \(\neq\) sin (A+B)

(B) A = \(30^\circ\), B = \(45^\circ\)

\(\therefore\) A + B = \(30^\circ\) +  \(45^\circ\) = \(75^\circ\)

Now, sin A + sin B = sin \(30^\circ\) + sin \(45^\circ\)

= \(\frac 12\) + \(\frac 1{\sqrt2}\) = \(\frac {1+\sqrt2}2 > 1\)

But sin (A+B) = sin \(75^\circ\) < 1 (\(\because\) -1 \(\leq\) sin \(\theta\) \(\leq\)1)

Hence, sin (A+B) \(\neq\) sin A + sin B

(C) A = \(45^\circ\), B = \(60^\circ\)

\(\therefore\) A + B = \(45^\circ\)+\(60^\circ\) = \(105^\circ\)

Nw, sin A + sin B = sin \(45^\circ\) + sin \(60^\circ\)

= \(\frac 1{\sqrt2}\) + \(\frac {\sqrt3}{2}\) = \(\frac {\sqrt2+\sqrt3}{2}\) >1

But sin (A+B) = sin \(105^\circ\) < 1 ( \(\because\) -1 \(\leq\) sin \(\theta\) \(\leq\) 1)

Hence, sin A + sin B \(\neq\) sin (A+B)

Correct option is: D) None of these