00:01
We want to prove that secant to the fourth theta minus tangent to the fourth theta equals secant squared theta plus tangent squared theta.
00:08
So, first of all, we can use the identity that the 1 plus tangent squared theta is equal to secant squared theta.
00:22
That's one of our trig identities that we're going to be using.
00:24
And on the left side of this equation, we're going to factor a difference of squares using the fact that if we have a difference of squares, a squared minus b squared, that is equal to a plus b times a minus b.
00:37
Or in this case, the a would be secant squared theta and the b would be tangent to the four, tangent squared theta.
00:49
Theta.
00:49
So if we factor this left side that would be secant squared theta plus tangent squared theta times secant squared theta minus tangent squared theta on the left side.
01:07
And if we take our identity that we wrote over here first and if we solve it for 1, if we subtract tangent squared from both sides of the equation.
01:21
Isolate the 1.
01:24
That's going to give us 1 equals secant squared theta minus tangent squared theta.
01:31
Then we can use this to substitute secant squared theta minus tangent squared theta is equal to 1.
01:36
That's exactly what we have right here.
01:37
So we're left with secant squared theta plus tangent squared theta and then that's just multiplied by 1 which we multiply that by 1 it's not going to change.
01:48
We get is secant squared theta plus tangent squared theta.
01:52
And that's exactly what we have on the right side of the equation, secant squared theta plus tangent squared theta.
01:58
So that proves that they're equivalent.
01:59
Now, in the second part of this video, we're going to actually use that identity to solve an equation, secant to the fourth theta equals tangent to fourth theta plus 3 tangent theta over the interval from x is greater than or equal to negative 180 degrees and less than or equal to 180 degrees.
02:18
Degrees.
02:18
So we can first see that our equation has a secant to the fourth power and a tangent to the fourth power and we already discovered that if we subtract those we get this.
02:33
So let's subtract tangent to the fourth theta from both sides and then that leaves us with secant to the fourth theta minus tangent to the fourth theta equals three tangent theta.
02:49
Theta and then using our identity we're going to replace secant fourth minus tangent fourth with secant squared plus tangent squared because they are equivalent so secant squared theta plus tangent squared theta since they are equivalent that's equal to three tangent theta and then we can get everything in terms of tangents again using the fact that secant squared theta is equal to one plus tangent squared theta it's just an identity so we can replace this secant squared theta with 1 plus tangent squared theta plus tangent squared theta equals 3 tangent theta.
03:32
We can combine these like terms that's going to be 1 plus 2 tangent squared theta equals 3 tangent theta and we could subtract the 3 tangent from both sides and rearrange this equation a bit into quadratic form.
03:48
We get 2 2 tangent squared theta minus 3 tangent theta plus 1 equals 0...