Question

Sin x / cos^2 (x/2)=4sin^2(x/2)

Answer 1

Все на фото. Используется sin 2x = 2 sin x cos x

Answer 2

To prove the equation $\frac{\sin x}{\cos^2(x/2)} = 4\sin^2(x/2)$, we can use the double-angle identities and the reciprocal identities for sine and cosine. Here's the step-by-step proof:

Step 1: Express $\cos^2(x/2)$ in terms of $\sin(x/2)$ using the identity $\cos^2\theta = 1 - \sin^2\theta$.

$\cos^2(x/2) = 1 - \sin^2(x/2)$

Step 2: Substitute the expression for $\cos^2(x/2)$ into the left side of the equation:

$\frac{\sin x}{1 - \sin^2(x/2)} = 4\sin^2(x/2)$

Step 3: Now, let's get rid of the fraction by multiplying both sides by $1 - \sin^2(x/2)$:

$\sin x = 4\sin^2(x/2) \cdot (1 - \sin^2(x/2))$

Step 4: Simplify the right side using the identity $(a^2 - b^2) = (a + b)(a - b)$:

$\sin x = 4\sin^2(x/2) \cdot (1 - \sin^2(x/2))$

$\sin x = 4\sin^2(x/2) \cdot \left(1 - \sin(x/2)\right) \cdot \left(1 + \sin(x/2)\right)$

Step 5: Apply the identity $\sin^2\theta = \frac{1}{2} - \frac{1}{2}\cos(2\theta)$:

$\sin x = 4 \left(\frac{1}{2} - \frac{1}{2}\cos(x)\right) \cdot \left(1 + \sin(x/2)\right)$

Step 6: Distribute and simplify further:

$\sin x = 2 - 2\cos(x) + 2\sin(x/2) - 2\sin(x/2)\cos(x)$

Step 7: Now, use the double-angle identity for sine: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$:

$\sin x = 2 - 2\cos(x) + 2\sin(x/2) - \sin(x)$

Step 8: Move the term $-\sin(x)$ to the left side:

$\sin x + \sin(x) = 2 - 2\cos(x) + 2\sin(x/2)$

Step 9: Combine like terms:

$2\sin x = 2 - 2\cos(x) + 2\sin(x/2)$

Step 10: Divide both sides by 2:

$\sin x = 1 - \cos(x) + \sin(x/2)$

Step 11: Finally, use the identity $\sin(x) = 2\sin(x/2)\cos(x/2)$:

$\sin x = 1 - \cos(x) + 2\sin(x/2)\cos(x/2)$

Step 12: Apply the double-angle identity for cosine: $\cos(2\theta) = 1 - 2\sin^2(\theta)$:

$\sin x = 1 - \cos(x) + 2\cdot\frac{1}{2}\cdot(1+\cos(x))$

Step 13: Simplify:

$\sin x = 1 - \cos(x) + 1 + \cos(x)$

Step 14: Combine like terms:

$\sin x = 2$

And there we have it: $\sin x = 2$.

However, it's important to note that there is a mistake in the proof, as the equation