(a) If $\tan(\alpha + i\beta) = x + iy$, then prove that (i) $x^2 + y…

Question

(a) If $\tan(\alpha + i\beta) = x + iy$, then prove that

(i) $x^2 + y^2 + 2x \cot 2\alpha = 1$, (ii) $x^2 + y^2 - 2y \coth 2\beta = -1$

(b) If $a + ib = c \tan(x + iy)$ then prove that $\tan 2x = \frac{2ac}{c^2 - a^2 - b^2}$.

(c) Sum the following series:

$$\sin \alpha \sin \beta + \sin 2\alpha \sin^2 \beta + \sin 3\alpha \sin^3 \beta + \dots$

Related pages

• Hyperbolic Functions of a Complex Variable — topic
• Engineering Mathematics-I syllabus

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