Question

Sin4x*tg2x+2cos^2 2x

Answer 1

$\sin 4x \cdot \text{tg} \ 2x + 2\cos^{2} 2x = 2\sin2x \cos 2x \cdot \dfrac{\sin 2x}{\cos 2x} + 2\cos^{2} 2x = \\= 2\sin^{2} 2x + 2\cos^{2} 2x = 2(\sin^{2} 2x + \cos^{2} 2x) = 2 \cdot 1 = 2$

Answer 2

To simplify the given expression, let's break it down step by step:

1. Start with the expression: sin(4x) * tan(2x) + 2cos^2(2x).

2. Recall the trigonometric identity: tan(x) = sin(x) / cos(x).

3. Substitute tan(2x) with sin(2x) / cos(2x): sin(4x) * (sin(2x) / cos(2x)) + 2cos^2(2x).

4. Multiply sin(4x) by sin(2x) and divide by cos(2x): (sin(4x) * sin(2x)) / cos(2x) + 2cos^2(2x).

5. Apply the double-angle formula for sin(2x): sin(4x) * (2sin(x)cos(x)) / cos(2x) + 2cos^2(2x).

6. Simplify the expression: 2sin(x)cos(x) * sin(4x) / cos(2x) + 2cos^2(2x).

7. Rewrite cos(2x) as 2cos^2(x) - 1: 2sin(x)cos(x) * sin(4x) / (2cos^2(x) - 1) + 2cos^2(2x).

8. Multiply 2sin(x)cos(x) with sin(4x) in the numerator: (2sin(x)cos(x) * sin(4x)) / (2cos^2(x) - 1) + 2cos^2(2x).

9. Apply the double-angle formula for cos(2x): (2sin(x)cos(x) * sin(4x)) / (2cos^2(x) - 1) + 2(2cos^2(x) - 1)^2.

10. Simplify the expression further if necessary, but this is the simplified form of the given expression.

Please note that without specific values for x, the expression cannot be further simplified numerically.