WEBVTT

1
00:00:00.000 --> 00:00:05.248
Here is the problem: 1 plus tan(x) squared equals sec(x) squared.

2
00:00:05.698 --> 00:00:07.532
Prove a trigonometric identity.

3
00:00:08.332 --> 00:00:12.721
Rewrite everything in terms of sin x and cos x. Identities used:.

4
00:00:13.171 --> 00:00:15.412
Left side in sine and cosine.

5
00:00:15.862 --> 00:00:18.068
Right side in sine and cosine.

6
00:00:18.518 --> 00:00:28.224
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
00:00:29.024 --> 00:00:33.633
Replace every cos squaredx with 1 minus sin squaredx and collect terms.

8
00:00:34.083 --> 00:00:44.056
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
00:00:44.856 --> 00:00:50.324
So the answer is The identity is true (for every value where both sides are defined).

10
00:00:50.774 --> 00:01:07.516
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 1 plus tan(x) squared equals sec(x) squared holds for every x at which both sides are defined.

11
00:01:08.316 --> 00:01:12.461
Numeric spot check at 10 sample values: both sides agree.