Trigonometry Identity Tan(squared)x+1=Sec(squared)x

We are talking about the trigonometric identities in particular to the Pythagorean identities for (tan) and (cot) of (theta), and (sec) and (cosec).0000

Here we are being asked to prove the identity (cot)² +1 = (cosec)².0012

The trick there is to remember the original Pythagorean identity.0020

Let me write that down to start with, sin²x + cos²x = 1, that is the original Pythagorean identity.0024

That should be very familiar to you because you use that all the time in trigonometry.0035

We will start with that and I know I’m trying to find (cot) and (cosec) in this somehow.0043

What I’m going to do is divide both sides by sin²x.0049

Divide everything here by sin²x and that gives me 1.0063

Now (cos)²/(sin)² that is (cos)/(sin)², that is (cot)²x.0072

1/(sin)² that is 1/(sin), that is (cosec) by definition.0081

There you have it, that is a pretty quick one.0090

We start with the original Pythagorean identity and we just divide both sides by (sin)²x to make it look exactly like what we are asked for.0093

I can rearrange terms here and I can get (cot)²x + 1 is equal to (cosec)²x.0104

This just comes back to knowing the original Pythagorean identity, if you remember that original Pythagorean identity.0119

And if you remember the definitions of sec, tan, cosec, and cot, then you can pretty quickly derive the new one cot² + 1 = cosec².0125

You might not even really need to memorize that one as long as you know the others vey well because you can work it out quickly.0136

Hi we are given (sec) of an angle here, 13/12 and we are given the angle in terms of degrees is between 270 and 360 and we are asked to find the tan(theta).0000

This is pretty clearly asking us to use the Pythagorean identity 10 ²(theta) +1= sec²(theta).0012

Sec²(theta) is we plug in 13/12, that is 13/12² and that is 169/144.0023

10²(theta) if we subtract 1 from both sides is 169/144 – 1 which is 169-144/144 which is 25/144.0038

Tan(theta) if we take the square root of both sides is + or – the square root of 25/144, which is + or -, those are both perfect squares so it comes out neatly 5/12.0067

Now the question is, whether it is the positive one or the negative one and we need to figure that out for the problem.0088

There is extra information given in the problem that we have not use yet.0094

We have not use the fact that (theta) is between 270 and 360 degrees .0098

Let me draw our (theta) would be approximately, here is my unit circle, in degrees that is 0, 90, 180, 270, and 360.0104

(theta) is between 270 and 360, (theta) is down there somewhere, so (theta) is an angle around there.0123

If you remember all students take calculus, that tells you which of these basic functions are positive and negative in each quadrants.0133

In the first quadrant they are all positive, in the second quadrant only (sin) is, third quadrant only (tan) is, fourth quadrant only (cos) is.0145

Our (theta) is in the fourth quadrant, let me write that as (theta) is in quadrant 4.0156

Tan(theta) only cos is positive there, tan is not, so tan(theta) is negative.0177

When we are choosing between these positive and negative square root here, we know we have to pick the negative one, -5/12.0188

There are really two steps to figuring out how to do this problem, one is to remember the Pythagorean identity tan²(theta) + 1=sec²(theta).0210

Then we take the value of sec(theta) that we are given, we plug it in, we work it down through and we try to solve for tan(theta) and we end up taking the square root.0221

We get plus or minus in there and we do not know what we want if positive or negative.0232

The second step there was to use the information about what quadrant (theta) is in and once we know that (theta) is in quadrant 4, we know that its tan it has to be negative.0237

That is from the all student take calculus rule and tan(theta) is negative, we know we need to take the negative square root.0253

That is how we get tan(theta) is equal to -5/12.0260

That is our lesson on the Pythagorean identity tan²(theta) + 1= sec²(theta).0265

These are the trigonometry lectures for www.educator.com.0271

OK, welcome back to the trigonometry lectures on educator.com, and today we're talking about the identity, tan²(x)+1=sec²(x).0000

You really want to think about this as a kind of companion identity to the main Pythagorean identity, which is that sin²(x)+cos²(x)=1.0010

Hopefully, you've really memorized this identity, sin²(x)+cos²(x)=1, that's one that comes up everywhere in trigonometry.0022

Then sort of the companion Pythagorean identity to that for secants and tangents is tan²(x)+1=sec²(x).0034

There's another related identity, essentially just the same for cotangents and cosecants, is that cot²(x)+1=csc²(x).0045

You may wonder how you'll remember this identities.0058

For example, how do you remember whether it's tan²+1=sec², or maybe it's the other way around, sec²+1=tan².0062

Well, really the answer to that lies in the exercises that we're about to do.0072

We'll show you how to derive those identities from the original one sin²+cos²=1.0077

As long as you can remember sin²+cos²=1, you'll learn how you can use that to figure out the others and you really won't have to work hard to remember this new Pythagorean identities.0085

Let's jump right into that.0099

The first example here is to prove the identity tan²(x)+1=sec²(x).0102

The trick there is to remember the original Pythagorean identity, which is cos²(x)+sin²(x)=1, that's the original Pythagorean identity.0109

That's one that you really should memorize and remember throughout all your work with trigonometry.0127

What you do to manipulate this into the new identity, is you just divide both sides by cos²(x).0136

On the left, we get cos²(x)/cos²(x) plus, let me write it as (sin(x)/cos(x)²=1/cos²(x).0152

Then of course, cos²(x)/cos²(x) is just 1, sin/cos, remember that's the definition of tangent, so this is tan²(x), 1/cos, that's the definition of sec(x), so sec²(x).0173

You can just rearrange this into tan²(x)+1=sec²(x).0191

That shows that this new identity is just a straight consequence of the original Pythagorean identity, so really if you can remember that Pythagorean identity, you can pretty quickly work out this new Pythagorean identity for tangents and cotangents.0203

Let's try using the Pythagorean identity for tangents and cotangents.0224

In this problem, we're given the tan(θ)=-4.21, and θ is between π/2 and π, we want to find secθ.0230

Remembering the Pythagorean identity, tan²(θ)+1=sec²(θ).0238

If we plug in what we're given here, tan²(θ), that's (-4.21²)+1=sec²(θ).0246

Well, 4.21², that's something I'm going to workout on my calculator, is 17.7241.0259

So, 17.7241+1=sec²(θ), that's 18.7241=sec²(θ).0274

If I take the square root of both sides, I get sec(θ) is equal to plus or minus the square root of 18.7241.0290

Again, I'm going to do that square root on the calculator, and I get approximately 4.33.0303

The question here is whether we want the positive or negative square root, and that's where we use the other piece of information given in the problem.0323

Θ is in the second quadrant here, θ is between π/2 and π, so θ is somewhere over here.0332

Now, sec(θ) remember, is 1/cos(θ).0348

In the second quadrant, if you remember All Students Take Calculus.0353

In the second quadrant, sine is positive, but cosine is not, cosine is negative.0358

That means, sec(θ) is negative.0365

θ is in quadrant 2, cos(θ) is negative, sec(θ), because that's 1/(θ), is negative.0370

So, sec(θ), we want to take the negative square root there, and we get an approximate value of -4.33.0394

That negative is very important.0411

The key to that problem is first of all invoking this Pythagorean identity, tan²+1=sec².0415

That's very important to remember.0423

We plug in the value that we're given and then we work it through and we'll try to solve for sec(θ).0426

We get this plus or minus at the end because we don't know if we want the positive square root or the negative square root.0431

That's where we use the fact that θ is in the second quadrant.0437

Since θ is in the second quadrant, cos(θ) must be negative, sec(θ), remember, is just 1/cos(θ), must also be negative.0448

That's how we know we want the negative square root there, so we get -4.33.0458

In our next example, we're given trigonometric identity and we're asked to prove it, (csc(θ)-cot(θ))/(sec(θ)-1) = cot(θ).0467

I'm going to start with the left-hand side of this trigonometric identity and I'm going to manipulate it.0480

I'm going to try and manipulate it into the right-hand side.0490

The left-hand side which is (csc(θ)-cot(θ))/(sec(θ)-1).0496

Now, I'm going to do something clever here and it's based on this old principle of whenever you have an (a-b) in the denominator, try multiplying by the conjugate, (a+b)/(a+b).0511

If it's the other way around, if you have (a+b), you multiply by the conjugate (a-b)/(a-b).0524

The reason you do that is that it makes the denominator (a²-b²).0530

We're taking advantage of that old formula from algebra, the difference of squares formula, and a lot of times the (a²-b²) is something that it'll simplify into something nice.0538

That's what's going to happen in here.0549

We're going to multiply, since we have (sec(θ)-1 in the denominator, by sec(θ)+1.0552

What that turns into in the denominator is this (a²-b²) form, so sec²(θ)-1.0565

The numerator really doesn't have anything very good happening yet, (csc(θ)-cot(θ))×sec(θ)+1, nothing very good happening in the numerator yet.0577

In the denominator, we're going to take advantage of the fact that tan²(θ)+1=sec²(θ), that's the Pythagorean identity that we're studying today.0593

If you move that around, you get sec²(θ)-1=tan²(θ).0607

That converts the denominator into tan²(θ), so that's very useful.0614

In the numerator, we have csc(θ)-cot(θ) and sec(θ)+1.0623

I think now I'm going to translate everything into sines and cosines because those will be easier to understand than cosecants and cotangents.0628

So, cosecant, remember is 1/sin(θ), minus cotangent is cos(θ)/sin(θ), sec(θ)+1, well, sec(θ) is 1/cos(θ), and tan(θ), if we translate into sines over cosines, that's sin²(θ)/cos²(θ).0637

Now, I've got fractions divided by fractions, I think the easiest thing to do here is to bring the denominator, flip it up and multiply it by the numerator.0670

We get cos²(θ)/sin²(θ), that's flipping up the denominator, and then the old numerator is, if I combined the first two terms of the first one, it's (1-cos(θ))/sin(θ).0681

In the second one, we have 1/cos(θ)+1.0706

Now, I think what I'm going to do is I'm going to look at this cosine squared, write it as cos(θ)×cos(θ).0715

Then pull one of those cosines over and multiply it by this term, and try to simplify things a little bit that way.0726

That will leave me with just one cosine left, still have sin² in the denominator, (1-cos(θ)/sin(θ)) times, now, cos(θ)×(1/cos(θ) is 1, plus 1×cos(θ).0736

Now, look, we have (1-cos(θ))×(1+cos(θ)).0767

We're going to use this pattern again, the (a-b)×(a+b)=a²-b².0772

If cos(θ)/sin²(θ), now multiply (1-cos)×(1+cos) gives us (1-cos²(θ)) and sin(θ) in the denominator.0780

Now, we're going to use the other Pythagorean identity, the original one which are right over here, sin²(θ)+cos²(θ)=1, if you switched that around 1-cos²(θ)=sin²(θ).0798

I'll plug that in, cos(θ)/sin(θ), 1-cos²(θ), now translates into sin²(θ) all divided by sin(θ).0819

I forgot my squared there, that's from the line above.0836

Now, we have some cancellations, sin²(θ) cancel, we get cos(θ)/sin(θ), but that's cot(θ).0840

By definition of cotangent, cotangent is defined to be cos/sin.0852

That's exactly equal to the right-hand side of the identity.0858

When you're proving this trigonometric identities, you pick one side and you multiply it as much as you can.0864

You try to manipulate it into the other side.0870

It does takes some trial and error.0874

I worked this problem out ahead of time, I tried a couple of different things and I finally found a way that gets us through it relatively quickly.0876

It's not like you'll always know exactly which path to follow, it takes a little bit of experimentation.0882

There are some patterns that you see over and over again, and the two patterns that we really invoke strongly in this one are these algebra pattern where (a-b)×(a+b)=a²-b².0888

Essentially, whenever you see an (a-b) or an (a+b), think about multiplying it by the conjugate, and then taking advantage of that pattern.0904

That's what really got us started on the first step here.0913

We had a sec(θ)-1, and I thought, okay, let's multiply that by sec(θ)+1, that gives us sec²-1.0917

The second pattern that we use there was to invoke these Pythagorean identities, tan²+1=sec², and sin²+cos²=1.0927

We invoked that one here, converting sec²-1 into tan².0942

Then later on, when he had another (a-b)×(a+b), it converted into 1-cos² and that in turn, by the Pythagorean identity converted into sin².0947

It's just a lot of manipulation but you can let your manipulation be kind of guided by these common algebraic identities and these common Pythagorean identities.0962

Then you just try to keep manipulating until you get to the other side of the equation.0973

We'll try some more examples later on.0978