To solve this equation, we can use trigonometric identities. First, let's simplify the equation:
7/4cos(x/4) = cos^3(x/4) + sin(x/2)
Since cos^3(x/4) is equal to (cos(x/4))^3, we can rewrite the equation as:
7/4cos(x/4) = (cos(x/4))^3 + sin(x/2)
Now, let's use the trigonometric identity: cos(2θ) = 1 - 2sin^2(θ)
By substituting θ = x/4, we get:
cos(x/2) = 1 - 2sin^2(x/4)
Rearranging the equation:
2sin^2(x/4) = 1 - cos(x/2)
Now, we substitute this back into our original equation:
7/4cos(x/4) = cos(x/4) * (1 - cos(x/2)) + sin(x/2)
Expanding the equation:
7/4cos(x/4) = cos(x/4) - cos(x/2)cos(x/4) + sin(x/2)
Rearrange the equation:
7/4cos(x/4) = cos(x/4) + sin(x/2) - (cos(x/2)sin(x/4))
Now, we can use the double angle identities:
sin(2θ) = 2sin(θ)cos(θ)cos(2θ) = cos^2(θ) - sin^2(θ)
Using these identities, we can simplify the equation further. Feel free to simplify the equation and solve for x.
To solve this equation, we can use trigonometric identities. First, let's simplify the equation:
7/4cos(x/4) = cos^3(x/4) + sin(x/2)
Since cos^3(x/4) is equal to (cos(x/4))^3, we can rewrite the equation as:
7/4cos(x/4) = (cos(x/4))^3 + sin(x/2)
Now, let's use the trigonometric identity: cos(2θ) = 1 - 2sin^2(θ)
By substituting θ = x/4, we get:
cos(x/2) = 1 - 2sin^2(x/4)
Rearranging the equation:
2sin^2(x/4) = 1 - cos(x/2)
Now, we substitute this back into our original equation:
7/4cos(x/4) = cos(x/4) * (1 - cos(x/2)) + sin(x/2)
Expanding the equation:
7/4cos(x/4) = cos(x/4) - cos(x/2)cos(x/4) + sin(x/2)
Rearrange the equation:
7/4cos(x/4) = cos(x/4) + sin(x/2) - (cos(x/2)sin(x/4))
Now, we can use the double angle identities:
sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos^2(θ) - sin^2(θ)
Using these identities, we can simplify the equation further. Feel free to simplify the equation and solve for x.