Question

If tan x = a and tan y = b
Show that sin(x+y) = (a+b) ÷ √(1+a^2)(1+b^2)

Answer 1

Answer on Question #62165 — Math – Trigonometry

Question

1. If $\tan x = a$ and $\tan y = b$.

Show that

$$\sin(x + y) = \frac{(a + b)}{\sqrt{(1 + a^2)(1 + b^2)}}.$$

Solution

Using

$$\sin(x + y) = \sin x \cos y + \cos x \sin y,$$ $$\tan x = \frac{\sin x}{\cos x} = a,$$ $$\tan y = \frac{\sin y}{\cos y} = b,$$ $$\sin^2 x + \cos^2 x = 1,$$ $$\sin^2 y + \cos^2 y = 1,$$

rewrite

$$\sin(x + y) = \sin x \cos y + \cos x \sin y = \frac{(\sin x \cos y + \cos x \sin y)}{\sqrt{((\cos x)^2 + (\sin x)^2)((\cos y)^2 + (\sin y)^2)}} = \frac{1}{2}$$ $$= \left| \text{dividing the numerator and the denominator by } \cos y \cos x \right| = \frac{\left(\frac{\sin x}{\cos x} + \frac{\sin y}{\cos y}\right)}{\sqrt{\left(1 + \left(\frac{\sin x}{\cos x}\right)^2\right) \left(1 + \left(\frac{\sin y}{\cos y}\right)^2\right)}} = \frac{\tan x + \tan y}{\sqrt{(1 + \tan^2 x)(1 + \tan^2 y)}} = \frac{a + b}{\sqrt{(1 + a^2)(1 + b^2)}}.$$

Q.E.D.

www.AssignmentExpert.com