Question

Sin^4 x+cos^4 x=cos^2 2x+1/4

Answer 1

Ответ:

$\frac{\pi }{8} +\frac{\pi n }{4} ,~n\in\mathbb {Z}$

Объяснение:

$sin^{4} x+cos^{4} x= cos^{2} 2x+\frac{1}{4} ;$

Воспользуемся формулами понижения степеней:

$sin^{2} x=\frac{1-cos2x}{2} ;cos^{2} x=\frac{1+cos2x}{2} .$

Получим

$(\frac{1-cos2x}{2} )^{2} +(\frac{1+cos2x}{2} ) ^{2} = cos^{2} 2x+\frac{1}{4} ;\\\\\frac{1-2cos2x+cos^{2} 2x}{4} +\frac{1+2cos2x+cos^{2}2x }{4} = cos^{2} 2x+\frac{1}{4}|*4 ;\\\\1-2cos2x+cos^{2} 2x+1+2cos2x+cos^{2} 2x= 4cos^{2} 2x+1;\\\\2+2cos^{2} 2x=4cos^{2} 2x+1;\\\\2cos^{2} 2x=1;\\1+cos4x=1;\\\\cos4x=0;\\\\4x=\frac{\pi }{2} +\pi n,~n\in\mathbb {Z}\\\\x=\frac{\pi }{8} +\frac{\pi n }{4} ,~n\in\mathbb {Z}$

Answer 2

To verify the given equation sin^4(x) + cos^4(x) = cos^2(2x) + 1/4, we can simplify both sides separately and check if they are equal.

Starting with the left-hand side (LHS): sin^4(x) + cos^4(x)

Using the identity cos^2(x) = 1 - sin^2(x), we can rewrite the expression as: (1 - cos^2(x))^2 + cos^4(x)

Expanding the squared term: (1 - 2cos^2(x) + cos^4(x)) + cos^4(x)

Simplifying: 1 - 2cos^2(x) + cos^4(x) + cos^4(x) 1 - 2cos^2(x) + 2cos^4(x)

Moving on to the right-hand side (RHS): cos^2(2x) + 1/4

Using the double-angle identity cos(2x) = 2cos^2(x) - 1, we can substitute it into the equation: cos^2(2x) = (2cos^2(x) - 1)^2 cos^2(2x) = 4cos^4(x) - 4cos^2(x) + 1

Substituting this into the RHS: 4cos^4(x) - 4cos^2(x) + 1 + 1/4 4cos^4(x) - 4cos^2(x) + 5/4

Now, let's compare the simplified expressions for LHS and RHS: LHS: 1 - 2cos^2(x) + 2cos^4(x) RHS: 4cos^4(x) - 4cos^2(x) + 5/4

As we can see, the expressions on the LHS and RHS are not equal. Therefore, the given equation sin^4(x) + cos^4(x) = cos^2(2x) + 1/4 is not true.