Question

Sin(x)^4+cosx^4=cos^2(2x)+1/4

Answer 1

Решение смотрите на фотографии

Answer 2

To verify the trigonometric identity $\sin(x)^4 + \cos(x)^4 = \cos^2(2x) + \frac{1}{4}$, we'll start by manipulating the right-hand side (RHS) to see if it matches the left-hand side (LHS) after simplification.

First, we'll simplify $\cos^2(2x)$: $\cos^2(2x) = \frac{1 + \cos(4x)}{2}$

Now, let's write the original identity again: $\sin(x)^4 + \cos(x)^4 = \frac{1 + \cos(4x)}{2} + \frac{1}{4}$

Next, we'll try to express $\sin(x)^4$ and $\cos(x)^4$ in terms of $\cos(4x)$. Using trigonometric identities, we know that: $\sin^2(x) = \frac{1 - \cos(2x)}{2}$ $\cos^2(x) = \frac{1 + \cos(2x)}{2}$

Now, to find $\sin(x)^4$ and $\cos(x)^4$, we square the expressions for $\sin^2(x)$ and $\cos^2(x)$: $\sin^4(x) = \left(\frac{1 - \cos(2x)}{2}\right)^2 = \frac{1 - 2\cos(2x) + \cos^2(2x)}{4}$ $\cos^4(x) = \left(\frac{1 + \cos(2x)}{2}\right)^2 = \frac{1 + 2\cos(2x) + \cos^2(2x)}{4}$

Now, we'll substitute these values back into the original identity: $\sin(x)^4 + \cos(x)^4 = \frac{1 - 2\cos(2x) + \cos^2(2x)}{4} + \frac{1 + 2\cos(2x) + \cos^2(2x)}{4}$

Notice that the $-2\cos(2x) + 2\cos(2x)$ terms cancel each other out, leaving us with: $\sin(x)^4 + \cos(x)^4 = \frac{1 + \cos^2(2x)}{2}$

Now, recall the expression for $\cos^2(2x)$ from earlier: $\cos^2(2x) = \frac{1 + \cos(4x)}{2}$

Substitute this back into the equation: $\sin(x)^4 + \cos(x)^4 = \frac{1 + \frac{1 + \cos(4x)}{2}}{2}$

To simplify further, combine the fractions: $\sin(x)^4 + \cos(x)^4 = \frac{2 + 1 + \cos(4x)}{4}$

Finally, simplify the numerator: $\sin(x)^4 + \cos(x)^4 = \frac{3 + \cos(4x)}{4}$