Question

Cos^2 (pi/8 + x) + cos^2 (pi/8 - x) = 3/2

Answer 1

Ответ:

Применяем формулы понижения степени и формулу суммы косинусов .

$cos^2(\dfrac{\pi }{8}+x)+cos^2(\dfrac{\pi}{8}-x)=\dfrac{3}{2}\\\\\\\dfrac{1+cos(\dfrac{\pi }{4}+2x)}{2}+\dfrac{1+cos(\dfrac{\pi }{4}-2x)}{2}=\dfrac{3}{2}\ \Big|\cdot 2\\\\\\1+cos(\dfrac{\pi }{4}+2x)+1+cos(\dfrac{\pi }{4}-2x)=3\\\\\\cos(\dfrac{\pi }{4}+2x)+cos(\dfrac{\pi }{4}-2x)=1\\\\2\cdot sin\dfrac{\pi }{4}\cdot sin2x=1\\\\\\2\cdot \dfrac{\sqrt2 }{2}\cdot sin2x=1\\\\sin2x=\dfrac{\sqrt2}{2}\\\\2x=(-1)^{n}\cdot \dfrac{\pi }{4}+\pi n\ ,\ n\in Z$

$x=(-1)^{n}\cdot \dfrac{\pi }{8}+\dfrac{\pi n}{2}\ ,\ n\in Z$  

Answer 2

To solve the equation $\cos^2\left(\frac{\pi}{8} + x\right) + \cos^2\left(\frac{\pi}{8} - x\right) = \frac{3}{2}$, we can use trigonometric identities to simplify the equation. One useful identity is the double angle formula for cosine:

$\cos(2\theta) = 2\cos^2(\theta) - 1$

We can use this identity to simplify the equation. First, rewrite the given equation with the double angle formula:

$\cos^2\left(\frac{\pi}{8} + x\right) + \cos^2\left(\frac{\pi}{8} - x\right) = \frac{3}{2}$

$2\cos^2\left(\frac{\pi}{8} + x\right) - 1 + 2\cos^2\left(\frac{\pi}{8} - x\right) - 1 = \frac{3}{2}$

Now, let's isolate the terms with the cosine squared expressions:

$2\cos^2\left(\frac{\pi}{8} + x\right) + 2\cos^2\left(\frac{\pi}{8} - x\right) = \frac{5}{2}$

Now, divide both sides of the equation by 2:

$\cos^2\left(\frac{\pi}{8} + x\right) + \cos^2\left(\frac{\pi}{8} - x\right) = \frac{5}{4}$

Now we have a simpler equation:

$\cos^2\left(\frac{\pi}{8} + x\right) + \cos^2\left(\frac{\pi}{8} - x\right) = \frac{5}{4}$

We've simplified the equation, but it may not have a straightforward analytical solution for $x$. You may need to use numerical methods or a graphing calculator to approximate the values of $x$ that satisfy this equation.