cos^3 = (1/2 cosx)(1+cos2x)
Please make the left match the right.
Compute for the third derivative of the given function for x = -5. The given angle is in radians.
y=12sinx+1x+2+2x
To make the left side match the right side:
cos^3x = (1/2 cosx)(1 + cos2x)
First, we can simplify the right side by distributing the (1/2 cosx) term:
cos^3x = (1/2 cosx)(1) + (1/2 cosx)(cos2x)
Simplifying further, we have:
cos^3x = (1/2 cosx) + (1/2 cosx)(cos2x)
Next, we can simplify the expression (cos2x) using a trigonometric identity:
cos 2x = 2cos^2x - 1
Substituting this into the expression, we get:
cos^3x = (1/2 cosx) + (1/2 cosx)(2cos^2x - 1)
Now, let's simplify further:
cos^3x = (1/2 cosx) + cos^2x - (1/2 cosx)
Combining like terms, we have:
cos^3x = cos^2x
So, the left side (cos^3x) now matches the right side (cos^2x).
To make the left side of the equation match the right side, we can use a trigonometric identity.
First, let's expand the right side of the equation using the double angle formula for cosine:
cos^3(x) = (1/2 cos(x))(1 + cos(2x))
Using the double angle formula cos(2x) = 2cos^2(x) - 1, we can substitute it in the equation:
cos^3(x) = (1/2 cos(x))(1 + 2cos^2(x) - 1)
Now, simplify the right side of the equation:
cos^3(x) = (1/2 cos(x))(2cos^2(x))
= cos(x) * cos^2(x)
= cos^3(x)
By simplifying the equation, we see that the left side of the equation already matches the right side. Therefore, the equation is already solved.
cos^3(x) = (1/2 cos(x))(1 + cos(2x)) is the simplified form of the given equation.
RS = (1/2 cosx)(1 + 2cos^2 x -1)
= cos^3 x
= LS , if you meant cos^2 x