UPSEE Solved Model Paper 6

For solve by using quotient rule, the result will be a complex form. So, for the simplicity we can take an easy method - substitution method. This is the complex number form, so better to substitute the value to an exponent form cos θ + i sin θ = $e^{i\theta}$ sin θ + i cos θ = $i\left[\frac{1}{1}\sin\theta + \cos\theta\right]$ = i [- i sin θ + cos θ] = i [cos θ - i sin θ] = $ie^{-i\theta}$ $\left(\frac{\cos\theta + i\sin\theta}{\sin\theta + i\cos\theta}\right)^4$ = $\left(\frac{e^{i\theta}}{i e^{-i\theta}}\right)^4$ = $\frac{1}{1}\left(e^{i8\theta}\right)$ = cos 8θ + i sin 8θ ∴ $\left(\frac{\cos\theta + i\sin\theta}{\sin\theta + i\cos\theta}\right)^4$ = cos 8θ + i sin 8θ