verify the identity
sin x(1 - 2 cos^2 x + cos^4 x) = sin^5 x
1-2cos^2 x looks like 2cos^2x-1 its just backwards. I am not sure where to start.
good observation
recall that cos 2A = 2cos^2 A - 1 = 1 - 2sin^2 A
so 1 - 2 cos^2 x = 2sin^2 x - 1
LS
=sin x(1 - 2 cos^2 x + cos^4 x)
= sinx(2sin^2 x - 1 + (cos^2x)(cos^2x)
= sinx( 2 sin^2x - 1 + (1-sin^2x)(1 -sin^2x))
= sinx(2sin^2x - 1 + 1 - 2sin^2x + sin^4x)
= sinx(sin^4x)
= sin^5 x
= RS
To verify the given identity, we can use trigonometric identities and simplify both sides of the equation. Let's break it down step by step:
1. Start with the left side of the equation: sin x(1 - 2 cos^2 x + cos^4 x)
2. Notice that the expression within the brackets looks similar to the identity for the cosine of double angle, which states: cos(2x) = 2cos^2(x) - 1.
3. Rearrange the expression within the brackets by substituting 2cos^2 x - 1 for 1 - 2 cos^2 x:
sin x(2cos^2 x - 1 + cos^4 x)
4. Now we can rewrite cos^4 x as (cos^2 x)^2:
sin x(2cos^2 x - 1 + (cos^2 x)^2)
5. Next, observe that sin^2 x is equal to 1 - cos^2 x according to the Pythagorean identity for sine and cosine:
sin^2 x = 1 - cos^2 x
Rearrange the equation to get:
cos^2 x = 1 - sin^2 x
6. Substitute this expression into the previous step:
sin x(2cos^2 x - 1 + (1 - sin^2 x)^2)
7. Expand the squared term:
sin x(2cos^2 x - 1 + 1 - 2sin^2 x + sin^4 x)
8. Combine like terms:
sin x(2cos^2 x - 2sin^2 x + sin^4 x)
9. Factor out a common factor of sin x:
sin x(2cos^2 x - 2sin^2 x + sin^4 x) = sin x(cos^2 x - sin^2 x)(cos^2 x + sin^2 x)
10. Recall the Pythagorean identity cos^2 x + sin^2 x = 1:
sin x(1 - sin^2 x - sin^2 x)(1)
11. Simplify:
sin x(1 - 2sin^2 x)(1) = sin x(1 - 2sin^2 x)
12. Finally, notice that 1 - 2sin^2 x is equal to sin^2 x according to the Pythagorean identity:
sin x(sin^2 x) = sin^3 x
Therefore, the left side and the right side of the equation are equal, verifying the given identity sin x(1 - 2 cos^2 x + cos^4 x) = sin^5 x.