Given series is
sin(p + x) sin(p + q) + x2/2! sin(p + 2q) + ....
We know that eiα = cos α + i sin α.
We develop new series
cos p + x cos(p + q) + x2/2! cos(p + 2q) + ... + i(sin p + x sin(p + q) + x2/2! sin(p + 2q)+....)
= eip + xei(p + q) + x2/2! ei(p + 2q)+.....
= eip(1 + xiq + x2/2! ei2q + ....)
= eip(1 + (x eiq) + \(\frac{(xe^{i2q})^2}{2!}\)+....)
= eip. e\(xe^{iq}\) (\(\because\) ex = 1 + x + x2/2! + .......)
= eip + xeiq
= eip + x(cos q + i sin q) (\(\therefore \) eiα = cos α + i sin α)
= excosq + i (p + x sin q)
= excosq. ei(p +x sin q) (\(\because\) ea + b = ea. eb)
= excosq (cos(p +sin q) + i sin (p + x sin q))
By comparing imaginary part we get
sin p + x sin (p + q) + x2/2! sin(p + 2q) = excosq sin(p + x sin q).
Hence, option (A) is correct.