Question

2(sin⁶x+cos⁶x)-3(sin⁴x+cos⁴x)=cos2x

Answer 1

$2(\sin^6x+\cos^6x)-3(\sin^4x+\cos^4x)=\cos2x$

Преобразуем степени синусов:

$2((\sin^2x)^3+\cos^6x)-3((\sin^2x)^2+\cos^4x)=\cos2x$

Воспользуемся основным тригонометрическим тождеством и формулой косинуса двойного угла:

$2((1-\cos^2x)^3+\cos^6x)-3((1-\cos^2x)^2+\cos^4x)=2\cos^2x-1$

Воспользуемся формулами куба разности и квадрата разности:

$2(1-3\cos^2x+3\cos^4x-\cos^6x+\cos^6x)-\\-3(1-2\cos^2x+\cos^4x+\cos^4x)=2\cos^2x-1$

Упростим выражения в скобках:

$2(1-3\cos^2x+3\cos^4x)-3(1-2\cos^2x+2\cos^4x)=2\cos^2x-1$

Раскроем скобки:

$2-6\cos^2x+6\cos^4x-3+6\cos^2x-6\cos^4x-2\cos^2x+1=0$

После приведения подобных получим:

$-2\cos^2x=0$

$\cos x=0$

$x=\dfrac{\pi}{2}+\pi n,\ n\in\mathbb{Z}$

Ответ: $\dfrac{\pi}{2}+\pi n,\ n\in\mathbb{Z}$

Answer 2

Let's simplify the given expression step by step:

2(sin⁶x + cos⁶x) - 3(sin⁴x + cos⁴x)

First, we can factor out a common term of (sin⁴x + cos⁴x) from both terms:

(sin⁴x + cos⁴x)(2(sin²x + cos²x) - 3)

Now, we know that sin²x + cos²x is equal to 1, as it's the fundamental trigonometric identity. So, we can replace sin²x + cos²x with 1:

(sin⁴x + cos⁴x)(2(1) - 3)

Now, simplify further:

(sin⁴x + cos⁴x)(2 - 3)

(sin⁴x + cos⁴x)(-1)

Now, we can use the trigonometric identity sin²x + cos²x = 1 to express sin⁴x + cos⁴x:

(sin²x + cos²x)² - 2(sin²x)(cos²x)

Now, let's replace sin²x with 1 - cos²x in the expression:

((1 - cos²x) + cos²x)² - 2(1 - cos²x)(cos²x)

Now, expand the square of a binomial:

(1 - cos²x + cos²x)² - 2(1 - cos²x)(cos²x)

Simplify further:

(1)² - 2(1 - cos²x)(cos²x)

1 - 2(1 - cos²x)(cos²x)

Now, let's expand the expression inside the parentheses:

1 - 2(cos²x - cos⁴x)

Now, distribute the -2 through the parentheses:

1 - 2cos²x + 2cos⁴x

Now, we have simplified the expression to:

2cos⁴x - 2cos²x + 1

This is the simplified form of the given expression, which is equal to cos(2x):

2cos²x - 2cos²x + 1 = cos(2x)

So, the original expression 2(sin⁶x + cos⁶x) - 3(sin⁴x + cos⁴x) is indeed equal to cos(2x).