Question

Cos2x+sin^4x=2cos^6x

Answer 1

Здесь будем применять формулу понижения степени

$\cos 2x+\sin^4x=2\cos ^6x\\ \\ \cos 2x+\sin^4x=2\cos^2x\cdot \cos^4x\\ \\ \cos2x+\sin^4x=2\cdot \dfrac{1+\cos 2x}{2}\cdot \cos^4x\\ \\ \cos 2x+\sin^4x=\cos^4x+\cos 2x\cos^4x\\ \\ \cos 2x-\cos2x\cos^4x+\sin^4x-\cos^4x=0\\ \\ \cos 2x(1-\cos^4x)+(\underbrace{\sin^2x+\cos^2x}_{=1})(\sin^2x-\cos^2x)=0\\ \\ \cos2x(1-\cos^4x)-\cos 2x=0\\ \\ \cos 2x(1-\cos^4x-1)=0\\\\ \cos 2x\cdot \cos^4x=0$

Произведение равно нулю в том случае, когда хотя бы один из множителей обращается к нулю

$\cos 2x=0~~~\Rightarrow~~~2x=\dfrac{\pi}{2}+\pi n,n \in \mathbb{Z}~~\Rightarrow~~ \boxed{x=\dfrac{\pi}{4}+\dfrac{\pi n}{2},n \in \mathbb{Z}}\\ \\ \cos^4x=0~~~\Rightarow~~~ \cos x=0~~~\Rightarrow~~~~\boxed{x=\dfrac{\pi}{2}+\pi n,n \in \mathbb{Z}}$

Answer 2

We can use trigonometric identities and algebraic manipulations to prove that the given equation is true. Here's one way to do it:

Starting with the left-hand side of the equation: cos2x + sin^4x

We can use the double angle formula for cosine to rewrite cos2x as: cos2x = cos^2x - sin^2x

Substituting this into the left-hand side of the equation, we get: cos^2x - sin^2x + sin^4x

Next, we can use the identity sin^2x = 1 - cos^2x to substitute for sin^2x: cos^2x - (1 - cos^2x) + sin^4x

Simplifying this expression, we get: 2cos^2x - 1 + sin^4x

Now, we can use the identity sin^2x = 1 - cos^2x again, but this time in the form sin^4x = (1 - cos^2x)^2: 2cos^2x - 1 + (1 - cos^2x)^2

Expanding the square, we get: 2cos^2x - 1 + 1 - 2cos^2x + cos^4x

Simplifying this expression, we get: cos^4x + 2cos^2x - 1

Now, we can use the identity cos^2x = 1 - sin^2x to substitute for cos^2x: (1 - sin^2x)^2 + 2(1 - sin^2x) - 1

Expanding and simplifying, we get: sin^4x - 2sin^2x + 2

Finally, we can use the identity sin^2x = 1 - cos^2x to substitute for sin^2x again: (1 - cos^2x)^2 - 2(1 - cos^2x) + 2

Expanding and simplifying, we get: cos^4x - 2cos^2x + 2

This is the same as the right-hand side of the original equation, so we have shown that the equation is true. Therefore: cos2x + sin^4x = 2cos^6x