Question

Cos⁴(a)*sin²(a)*cos²(a); tg(a)=2​

Answer 1

Ответ:

cos⁴a + sin²a * cos²a = cos²a(cos²a + sin²a) = cos²a = 1/(1 / cos²a) = 1 / (1 + tg²a)

tga = 2, 1 + tg²a = 5

cos⁴a + sin²a * cos²a = 1/5

Answer 2

To simplify the expression $\cos^4(a) \cdot \sin^2(a) \cdot \cos^2(a)$ given that $\tan(a) = 2$, we can first express $\sin(a)$ and $\cos(a)$ in terms of $\tan(a)$:

1. Recall that $\tan(a) = \frac{\sin(a)}{\cos(a)}$.

2. We are given that $\tan(a) = 2$, so $\frac{\sin(a)}{\cos(a)} = 2$.

Now, we can solve for $\sin(a)$ and $\cos(a)$ in terms of $\tan(a)$:

$\sin(a) = 2\cos(a)$

Now, substitute this into the expression:

$$\begin{align*} \cos^4(a) \cdot \sin^2(a) \cdot \cos^2(a) &= \cos^4(a) \cdot (2\cos(a))^2 \cdot \cos^2(a) \\ &= 4\cos^6(a) \cdot \cos^2(a) \\ &= 4\cos^8(a) \end{align*}$$

So, the simplified expression $\cos^4(a) \cdot \sin^2(a) \cdot \cos^2(a)$ in terms of $\tan(a) = 2$ is $4\cos^8(a)$.