(a) Express each of the following in modulus-argument form, where $0<\theta<\frac{\pi}{2}$:
$$\text{i)}\ 1+i\tan\theta,\quad\text{ii)}\ 1+i\cot\theta,\quad\text{iii)}\ \frac{1}{\sin\theta}+\frac{1}{\cos\theta}i.$$
(b) Hence simplify each of the following:
$$\text{i)}\ (1+i\tan\theta)^2,\quad\text{ii)}\ (1+i\cot\theta)^{-3},\quad\text{iii)}\ \frac{1}{\sin\theta}-\frac{1}{\cos\theta}i.$$
I can answer all the questions in my math textbook except for this question. And there are no specific examples of this question in my textbook. Any help is appreciated.
THANKS
Best Answer
Most of your examples are of the form $$a(\cos\theta+i\sin\theta)$$ For example, $$1+i\tan\theta=\frac{1}{\cos\theta}(\cos\theta+i\sin\theta)$$ Can you do the rest? Please tell me if you need any more help.
Note that as $0<\theta<\frac{\pi}{2}$ the values of both $\sin\theta$ and $\cos\theta$ will always be positive.