Cos^4(x)+sin^4(4)=cos(2x)

Plz help.

I suspect a typo

http://www.wolframalpha.com/input/?i=Cos%5E4(x)%2Bsin%5E4(4)%3Dcos(2x)

notice the graph of the left side does not intersect the graph of the right side

I think you misplaced a minus sign. Fix it and think "difference of squares"

To solve the equation cos^4(x) + sin^4(4) = cos(2x), we need to apply some trigonometric identities and simplifications.

Let's break it down step by step:

Step 1: Start with the left-hand side (LHS) of the equation:
cos^4(x) + sin^4(4)

Step 2: Recall the identity for the fourth power of cosine:
cos^4(x) = (cos^2(x))^2

Step 3: Rewrite the equation using the identity from step 2:
(cos^2(x))^2 + sin^4(4)

Step 4: Expand the expression on the left-hand side to simplify further:
(cos^2(x))^2 = (cos^2(x)) * (cos^2(x)) = cos^2(x)*cos^2(x) = cos^2(x) * cos^2(x) = cos^2(x) * cos^2(x) = (cos^2(x))^2

Step 5: Apply the Pythagorean identity:
cos^2(x) + sin^2(x) = 1

Step 6: Substitute the Pythagorean identity into the expression from step 4:
(cos^2(x))^2 = (1 - sin^2(x))^2

Step 7: Simplify the expression:
(1 - sin^2(x))^2 = (1 - sin^2(x)) * (1 - sin^2(x)) = 1 - 2sin^2(x) + sin^4(x)

Substituting this simplified form back into the original expression, we have:

cos^4(x) + sin^4(4) = 1 - 2sin^2(4) + sin^4(4)

Step 8: Now, let's focus on the right-hand side (RHS) of the equation:
cos(2x)

Step 9: Recall the double angle identity for cosine:
cos(2x) = 2cos^2(x) - 1

Step 10: Substitute the expressions derived in Step 7 and Step 9 back into the original equation:

1 - 2sin^2(4) + sin^4(4) = 2cos^2(x) - 1

Step 11: Simplify the equation further:

1 - 2sin^2(4) + sin^4(4) = 2cos^2(x) - 1
1 - 2sin^2(4) + sin^4(4) + 1 = 2cos^2(x)
2 - 2sin^2(4) + sin^4(4) = 2cos^2(x)

Step 12: Apply another trigonometric identity:
sin^2(θ) = 1 - cos^2(θ)

Substituting this into the equation, we get:

2 - 2(1 - cos^2(4)) + (1 - cos^2(4))^2 = 2cos^2(x)

Step 13: Simplify the equation further:

2 - 2 + 2cos^2(4) + 1 - 2cos^2(4) + cos^4(4) = 2cos^2(x)
1 + cos^4(4) = 2cos^2(x)

Now we have the simplified equation:

1 + cos^4(4) = 2cos^2(x)

To continue solving for x, we need the specific value of cos(4). Please provide that value in order to proceed.