To solve this equation for x, we need to simplify the left side first using trigonometric identities.
Cos(2x) can be expanded using the double angle formula: cos(2x) = cos^2(x) - sin^2(x Sin(2x) can be expanded using the double angle formula: sin(2x) = 2sin(x)cos(x)
Now we rewrite the equation using the trigonometric identities:
cos^2(x) - sin^2(x) + 2sin(x)cos(x)cos(2x) = 7
Now we substitute cos^2(x) = 1 - sin^2(x) into the equation:
To solve this equation for x, we need to simplify the left side first using trigonometric identities.
Cos(2x) can be expanded using the double angle formula: cos(2x) = cos^2(x) - sin^2(x
Sin(2x) can be expanded using the double angle formula: sin(2x) = 2sin(x)cos(x)
Now we rewrite the equation using the trigonometric identities:
cos^2(x) - sin^2(x) + 2sin(x)cos(x)cos(2x) = 7
Now we substitute cos^2(x) = 1 - sin^2(x) into the equation:
1 - sin^2(x) - sin^2(x) + 2sin(x)cos(x)cos(2x) = 7
Combine like terms:
1 - 2sin^2(x) + 2sin(x)cos(x)cos(2x) = 7
Rearrange the equation:
-2sin^2(x) + 2sin(x)cos(x)cos(2x) = 6
Divide by 2:
-sin^2(x) + sin(x)cos(x)cos(2x) = 3
We can further simplify this equation by recognizing that sin(x)cos(x) = 1/2sin(2x):
-sin^2(x) + 1/2sin(2x)cos(2x) = 3
Since sin(2x) = 2sin(x)cos(x), we can substitute this into the equation:
-sin^2(x) + 1/2 * 2sin(x)cos(x) = 3
-sin^2(x) + sin(x)cos(x) = 3
Now we solve this trigonometric equation to find the possible values of x.