WEBVTT

1
00:00:00.000 --> 00:00:05.201
Here is the problem: cos(2x) equals 1 minus 2sin(x) squared.

2
00:00:05.651 --> 00:00:07.543
Prove a trigonometric identity.

3
00:00:08.343 --> 00:00:12.592
Rewrite everything in terms of sin x and cos x. Identities used:.

4
00:00:13.042 --> 00:00:15.341
Left side in sine and cosine.

5
00:00:15.791 --> 00:00:18.043
Right side in sine and cosine.

6
00:00:18.493 --> 00:00:28.257
Subtract the sides and write the difference over a common denominator. The identity is true exactly when this numerator is zero for every value of the variable.

7
00:00:29.057 --> 00:00:33.527
Replace every cos squaredx with 1 minus sin squaredx and collect terms.

8
00:00:33.977 --> 00:00:43.973
The numerator is identically zero. Every term cancelled after using sin squared plus cos squared equals 1, so both sides are equal wherever they are defined.

9
00:00:44.773 --> 00:00:49.951
So the answer is The identity is true (for every value where both sides are defined).

10
00:00:50.401 --> 00:01:07.572
After rewriting both sides with sine and cosine and replacing cos squared with 1 minus sin squared, the difference of the two sides is exactly 0, so cos(2x) equals 1 minus 2sin(x) squared holds for every x at which both sides are defined.

11
00:01:08.372 --> 00:01:12.563
Numeric spot check at 10 sample values: both sides agree.