How does one obtain cos2x = cos^2(x) - sin^2(x) by differentiating the identity sin2x = 2sinxcosx
taking derivative of
sin2x = 2sinxcosx with respect to x
2cos2x = (2sinx)(-sinx) + cosx(2cosx)
cos2x = cos^2x - sin^2x
and there is your result.
To obtain the identity cos2x = cos^2(x) - sin^2(x) from the identity sin2x = 2sin(x)cos(x) by differentiation, we need to use the differentiation rules for trigonometric functions.
First, let's differentiate sin2x = 2sin(x)cos(x) with respect to x.
Using the chain rule, we can rewrite sin2x as sin(u) and cos(x) as cos(v), where u = 2x and v = x.
So, differentiating sin2x = 2sin(x)cos(x) gives us:
cos(u) * du/dx = 2cos(v) * sin(v) + 2sin(v) * (-sin(v)),
where du/dx represents the derivative of u with respect to x.
Simplifying the right-hand side, we have:
cos(2x) * 2 = 2cos(x) * sin(x) - 2sin(x) * sin(x).
Now, let's simplify both sides of the equation.
Dividing both sides by 2, we get:
cos(2x) = cos(x) * sin(x) - sin(x) * sin(x).
Remembering the trigonometric identity sin^2(x) = 1 - cos^2(x), we can rewrite sin(x) * sin(x) as 1 - cos^2(x).
Substituting the identity into the equation, we have:
cos(2x) = cos(x) * (1 - cos^2(x)) - (1 - cos^2(x)).
Expanding the equation, we get:
cos(2x) = cos(x) - cos^3(x) - 1 + cos^2(x).
Finally, rearranging the terms and combining like terms, we obtain:
cos(2x) = cos^2(x) - sin^2(x).
Therefore, we have successfully obtained cos2x = cos^2(x) - sin^2(x) by differentiating the identity sin2x = 2sin(x)cos(x) and using trigonometric identities.