Question

Cos(п/3+в)*cos(п/3-в)

Answer 1

Ответ:

$\cos( \frac{\pi}{3} + \beta ) \times \cos( \frac{\pi}{3} - \beta ) = \frac{ \cos(2 \beta ) }{2} - \frac{1}{4}$

Answer 2

To simplify the expression $\cos\left(\frac{\pi}{3} + v\right) \times \cos\left(\frac{\pi}{3} - v\right)$, we'll use the cosine of the sum and difference of angles formulas:

1. Cosine of the sum of angles formula: $\cos(a + b) = \cos a \cos b - \sin a \sin b$

2. Cosine of the difference of angles formula: $\cos(a - b) = \cos a \cos b + \sin a \sin b$

Given that $a = \frac{\pi}{3}$ and $b = v$, we can rewrite the expression as follows:

$\cos\left(\frac{\pi}{3} + v\right) \times \cos\left(\frac{\pi}{3} - v\right)$ $= \left(\cos\left(\frac{\pi}{3}\right) \cos v - \sin\left(\frac{\pi}{3}\right) \sin v\right) \times \left(\cos\left(\frac{\pi}{3}\right) \cos(-v) + \sin\left(\frac{\pi}{3}\right) \sin(-v)\right)$ $= \left(\frac{1}{2} \cos v - \frac{\sqrt{3}}{2} \sin v\right) \times \left(\frac{1}{2} \cos v + \frac{\sqrt{3}}{2} \sin v\right)$

Now, let's expand and simplify further:

$= \frac{1}{4} \cos^2 v - \frac{3}{4} \sin^2 v$

This is the simplified expression for $\cos\left(\frac{\pi}{3} + v\right) \times \cos\left(\frac{\pi}{3} - v\right)$ in terms of $v$.