Please help me proof the identity. This isn't even math to me, I can not understand it so please show all work. God bless
A)1-cos2x/sin2x=tanx
B) sin4x/sinx=4cosxcos2x
If this isn't even math to you, you clearly haven't been paying attention during the discussion of the double-angle formulas ...
(1-cos2x)/sin2x
= (1-(2cos^2x-1))/(2sinx cosx)
= (2-2cos^2x)/(2sinx cosx)
= 2sin^2x/(2sinx cosx)
= sinx/cosx
= tanx
sin4x/sinx
= (2sin2x cos2x)/sinx
= (2*2sinx cosx cos2x)/sinx
= 4cosx cos2x
Sorry steve not everyone is as smart as you. Thanks for your help tho
Sure, I'd be happy to help you prove these identities. Let's start with identity A:
A) To prove the identity (1 - cos(2x)) / sin(2x) = tan(x), we'll need to work on the left-hand side of the equation until it matches the right-hand side.
1. Start with the left-hand side of the equation: (1 - cos(2x)) / sin(2x).
2. The first step is to simplify the numerator. Expand the expression 1 - cos(2x). By using the identity cos(2x) = cos^2(x) - sin^2(x), we can rewrite the expression as 1 - (cos^2(x) - sin^2(x)).
3. Simplify further by distributing the negative sign: 1 - cos^2(x) + sin^2(x).
4. Apply the Pythagorean identity sin^2(x) + cos^2(x) = 1 to simplify the expression: sin^2(x) - cos^2(x) + 1.
5. Now, let's work on the denominator. We have sin(2x) in the current expression, and we know that sin(2x) = 2sin(x)cos(x). Substitute this value into the denominator: 2sin(x)cos(x).
6. At this point, we have (sin^2(x) - cos^2(x) + 1) / (2sin(x)cos(x)).
7. To simplify this further, we can rewrite sin^2(x) - cos^2(x) as -(cos^2(x) - sin^2(x)). Recall that -(a - b) = b - a. So, we have (sin^2(x) - cos^2(x) + 1) / (2sin(x)cos(x)) = (cos^2(x) - sin^2(x) + 1) / (2sin(x)cos(x)).
8. Now, notice that the numerator cos^2(x) - sin^2(x) + 1 can be rearranged as sin^2(x) - (-cos^2(x)) + 1.
9. We can rewrite -cos^2(x) as -(cos(x))^2. Now, the numerator becomes sin^2(x) + (cos(x))^2 + 1.
10. Remember the Pythagorean identity sin^2(x) + (cos(x))^2 = 1. Substitute this value into the numerator: 1 + 1 + 1.
11. Simplify further: 3.
12. Now, the left-hand side (3) / (2sin(x)cos(x)).
13. Finally, we know that tan(x) = sin(x) / cos(x). Rewrite the left-hand side using tangent: 3 / (2sin(x)cos(x)) = 3 / (2tan(x)).
14. This matches the right-hand side of the equation, which is tan(x).
Therefore, we have proven the identity (1 - cos(2x)) / sin(2x) = tan(x).
Now, let's move on to prove identity B:
B) To prove the identity sin(4x) / sin(x) = 4cos(x)cos(2x):
1. Begin with the left-hand side: sin(4x) / sin(x).
2. Recall the double-angle identity for sine: sin(2x) = 2sin(x)cos(x). Using this identity, we can rewrite sin(4x) as 2sin(2x)cos(2x).
3. Now, substitute this value into the numerator of the left-hand side: 2sin(2x)cos(2x) / sin(x).
4. Using the double-angle identity again, we can rewrite sin(2x) as 2sin(x)cos(x).
5. Substituting this into the numerator: 2 * 2sin(x)cos(x)cos(2x) / sin(x).
6. Simplify further: 4sin(x)cos(x)cos(2x) / sin(x).
7. Notice that sin(x) cancels out in the numerator and denominator, leaving us with 4cos(x)cos(2x).
This matches the right-hand side of the equation, 4cos(x)cos(2x), proving the identity sin(4x) / sin(x) = 4cos(x)cos(2x).
I hope that helps you understand the process of proving these trigonometric identities! Let me know if you have any further questions.