Question

√2 cosx-√2sinx=1 Help me

Answer 1

Ответ

пошаговое объяснение:

Answer 2

To solve the equation $\sqrt{2} \cos(x) - \sqrt{2} \sin(x) = 1$, we can use trigonometric identities and algebraic manipulation. Here's a step-by-step solution:

1. Isolate trigonometric functions: Rewrite the equation in terms of sine and cosine functions:

   $$\sqrt{2} \cos(x) - \sqrt{2} \sin(x) = 1 \\ \Rightarrow \sqrt{2} (\cos(x) - \sin(x)) = 1 \\ \Rightarrow \cos(x) - \sin(x) = \frac{1}{\sqrt{2}}$$

2. Use trigonometric identity: Use the trigonometric identity $\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)$ to rewrite $\cos(x) - \sin(x)$ in a more manageable form:

   $$\cos(x) - \sin(x) = \frac{1}{\sqrt{2}} \\ \Rightarrow \cos(x) - \sin(x) = \cos\left(\frac{\pi}{4}\right) \\ \Rightarrow \cos\left(x - \frac{\pi}{4}\right) = \cos(x)\cos\left(\frac{\pi}{4}\right) + \sin(x)\sin\left(\frac{\pi}{4}\right)$$

3. Apply the trigonometric identity: Use the trigonometric identity $\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$ and $\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$ to rewrite the equation further:

   $$\cos\left(x - \frac{\pi}{4}\right) = \frac{\cos(x)}{\sqrt{2}} + \frac{\sin(x)}{\sqrt{2}}$$

4. Solve for $x$: Now, we can set $\frac{\cos(x)}{\sqrt{2}} + \frac{\sin(x)}{\sqrt{2}} = \frac{1}{\sqrt{2}}$:

   $$\frac{\cos(x) + \sin(x)}{\sqrt{2}} = \frac{1}{\sqrt{2}} \\ \Rightarrow \cos(x) + \sin(x) = 1$$

5. Use trigonometric identity: Use the trigonometric identity $\sin(\alpha) = \sqrt{1 - \cos^2(\alpha)}$ to express 0 0