in this video we're going to talk about how to simplify trigonometric expressions so before we work on this problem you may want to write down a few formulas to keep in mind so the first one will be the reciprocal identities secant is equal to 1 over cosine theta and cosecant theta is equal to 1 over sine theta now you also need to know that tangent is equal to sine theta divided by cosine theta and cotangent theta is equal to cosine over sine so make sure to write these formulas down because we're going to be using it throughout the course of this video and then you need to know the trigonometric pythagorean identities the first one sine squared plus cosine squared theta is equal to one and then we have one plus tangent squared is equal to secant squared and then one plus cotangent squared that's equal to cosecant squared theta now let's go ahead and get started with this one so we can replace secant with one over cosine theta sine theta we can leave it the way it is we can just write that as sine theta over one and then i'm going to rewrite cosine theta now all we could do at this point is we can cancel cosine theta so the final answer is simply sine theta so this entire expression can be simplified to sine theta now let's try this one tangent squared theta plus sine squared theta plus cosine squared theta go ahead and simplify this expression now if you recall one of the pythagorean theorems that i mean pythagorean identities that was done earlier is this one sine squared plus cosine squared is equal to one so what we're going to do is we're going to replace this part with 1. so we're going to have tangent squared theta plus 1. now if you recall another pythagorean identity is one plus tangent squared is secant squared so this entire expression here is equivalent to secant squared theta now let's move on to the next one sine squared times cosecant theta times secant theta how can we simplify this expression cosecant squared we know it's equivalent to one over sine and secant is one over cosine theta now sine squared we can write that as sine theta times sine theta and so we could cancel one of the sine thetas on top with the other one on the bottom and what we have left over is sine over cosine sine theta over cosine theta we can reduce that to or convert it to tangent theta and so tangent theta is equal to the entire original expression now let's move on to this one cotangent times tan plus cotangent what will this expression simplify to now we know that cotangent is well actually first before we change cotan into cosine and sine let's distribute so cotangent times tangent we'll just leave it as cotan tan and then cotangent times itself that's going to be cotangent squared theta now let's convert cotangent into cosine over sine and tangent is sine over cosine now i'm going to leave cotangent squared the way it is notice that sine cancels and cosine cancel sine divided by sine is one cosine divided by cosine is one so if this part is one then that is one one times one is going to be one so this whole thing reduces to one so we're left with one plus cotangent squared which is another pythagorean identity and as we wrote earlier in the beginning of the video one plus cotangent squared is equal to cosecant squared so the original expression can be reduced to cosecant square theta number five secant squared times one minus sine squared what can we do with this one now let's focus on the part one minus sine squared because that's related to a trigonometric identity we know that sine squared plus cosine squared is equal to one so we can replace one with what we have here or we could just replace this whole thing with cosine squared no matter how we do it the result will be the same so let's try replacing one with what we have here so this is going to be sine squared plus cosine squared minus sine squared so all we did was we replaced one with sine squared plus cosine squared now sine squared minus square sine squared that's going to cancel and so we're going to have secant squared times cosine squared theta now secant squared is 1 over cosine squared and then when you multiply that by cosine squared these two will cancel which means the final answer can be reduced to one so that's it for number five number six cosine theta plus sine theta times tangent theta how can we simplify this to a more simpler expression one of the first things we can do is we can replace tangent with sine over cosine now what do you think we need to do next because it doesn't look like anything simplifies right now one thing we could do is we can convert this into a single expression by turning this into a fraction and trying to get common denominators so i'm going to multiply cosine theta over 1 by cosine over cosine cosine over cosine is 1 so it doesn't change the value of cosine theta 1 times anything is going to be 1 times cosine is just cosine so cosine times cosine that's going to be cosine squared and cosine times 1 is simply cosine and for the other fraction we have sine times sine which is sine squared over cosine so now we have two fractions with the same denominator which means we can combine this into a single fraction by adding the numerators of the two fractions so we can write this as cosine squared plus sine squared theta all divided by cosine so now we have a pythagorean identity we know that cosine squared plus sine squared that's one and one over cosine that's a reciprocal identity that's secant so the final answer of the original expression is simply secant theta number seven feel free to pause the video and try this one so let's begin by using the reciprocal identities so we can convert secant into one over cosine sine theta we can just leave it the way it is for now cotangent we can write that as cosine over sine and tangent i'm going to replace that with sine over cosine now the next thing i'm going to do is i'm going to distribute the sine theta so sine times cosine over sine sine will cancel so we're still going to have one over cosine on top we're no longer going to have the brackets when sine and and when these two sines cancel we're just going to have cosine next sine theta times this fraction nothing's going to cancel so we're just going to have sine times sine which is sine squared and we have a cosine on the bottom notice we have a complex fraction we have fractions within a larger fraction so i'm going to try to get rid of these two fractions and notice that they have a similar denominator cosine so what i'm going to do is the big fraction i'm going to multiply the top of the big fraction by cosine and the bottom by cosine as well because cosine over cosine is one so if you multiply this fraction by one you're not changing the value of that fraction these will cancel and so i'm going to get a one on top here i have cosine times cosine which becomes cosine squared and i need to multiply this cosine by the other fraction and these two will cancel leaving behind sine squared so now we have a pythagorean identity on the denominator of this of this fraction so cosine squared plus sine squared we know it's one and one divided by one is simply one so this entire expression reduces to one so that's basically it for this video so now you know how to employ the reciprocal identities and the pythagorean identities to simplify trigonometric expressions