verify the identity sin(4u)=2sin(2u)cos(2u)
LS
=sin(4u)
= sin(2u + 2u)
= sin(2u)cos(2u) + cos(2u)sin(2u)
= 2sin(2u)cos(2u)
= RS
Good
To verify the given identity sin(4u) = 2sin(2u)cos(2u), we'll use a trigonometric identity called the double-angle identity for sine.
The double-angle identity for sine states that sin(2x) = 2sin(x)cos(x).
Using the double-angle identity, we can rewrite the right side of the given equation:
2sin(2u)cos(2u) = 2 * (2sin(u)cos(u)) * (cos^2(u) - sin^2(u))
= 4sin(u)cos(u)(cos^2(u) - sin^2(u))
Now, let's simplify the right side further:
4sin(u)cos(u)(cos^2(u) - sin^2(u)) = 4sin(u)cos(u)(cos^2(u) - (1 - cos^2(u)))
= 4sin(u)cos(u)(cos^2(u) - 1 + cos^2(u))
= 4sin(u)cos(u)(2cos^2(u) - 1)
Finally, applying the double-angle identity for sine to the expression obtained above:
4sin(u)cos(u)(2cos^2(u) - 1) = 4sin(u)cos(u)(2 * (cos(u))^2 - 1) = 8sin(u)cos(u)(cos(u))^2 - 4sin(u)cos(u)
Therefore, we have shown the right side equals:
2sin(2u)cos(2u) = 8sin(u)cos(u)(cos(u))^2 - 4sin(u)cos(u)
Comparing this to the left side of the given equation:
sin(4u) = 8sin(u)cos(u)(cos(u))^2 - 4sin(u)cos(u)
Since the right side of the equation matches with the left side, we have verified that sin(4u) = 2sin(2u)cos(2u).
To verify the identity sin(4u) = 2sin(2u)cos(2u), we will use the double angle formulas for sine and cosine.
The double angle formulas for sine and cosine are as follows:
sin(2u) = 2sin(u)cos(u)
cos(2u) = cos^2(u) - sin^2(u)
Now, let's substitute these formulas into the given identity:
sin(4u) = 2sin(2u)cos(2u)
Using the double angle formulas, we substitute sin(2u) and cos(2u) in this equation:
sin(4u) = 2(2sin(u)cos(u))(cos^2(u) - sin^2(u))
Expanding this equation further, we use the distributive property:
sin(4u) = 4sin(u)cos(u)(cos^2(u) - sin^2(u))
Now, let's simplify the expression inside the parentheses:
cos^2(u) - sin^2(u) = cos^2(u) - (1 - cos^2(u))
cos^2(u) - sin^2(u) = cos^2(u) - 1 + cos^2(u)
cos^2(u) - sin^2(u) = 2cos^2(u) - 1
Substituting this simplified expression back into the equation, we have:
sin(4u) = 4sin(u)cos(u)(2cos^2(u) - 1)
Now, let's simplify the expression further:
sin(4u) = 8sin(u)cos^3(u) - 4sin(u)cos(u)
At this point, we can simplify the right side of the equation:
8sin(u)cos^3(u) - 4sin(u)cos(u) = 4sin(u)cos(u)(2cos^2(u) - 1)
Finally, we have shown that sin(4u) = 2sin(2u)cos(2u), and therefore, the given identity is verified.