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TRIGONOMETRIC IDENTIES AND EQUATIONS Double-angle identities: Problem type 1Find sin2x, cos2x, and tan2x if tanx =and x terminates in quadrant II:sin 2xX 52cos 2xta...

Question

TRIGONOMETRIC IDENTIES AND EQUATIONS Double-angle identities: Problem type 1Find sin2x, cos2x, and tan2x if tanx =and x terminates in quadrant II:sin 2xX 52cos 2xtan 2x

TRIGONOMETRIC IDENTIES AND EQUATIONS Double-angle identities: Problem type 1 Find sin2x, cos2x, and tan2x if tanx = and x terminates in quadrant II: sin 2x X 5 2 cos 2x tan 2x



Answers

Solve the given problems. Find the exact value of $\cos 2 x+\sin 2 x \tan x$

Okay, so we're given that sequence of X, which is actually, um, sequence of X is equal to one over co sign of ex service is equal to the square roots of 5/1. So we see that if you draw a triangle co sign of X is equal to one over the square root of five services one square with five year and all its solved for why value or a sign about build. So this is the square root of five squared minus one squared is equal to well thought his question work squared. So we get five minus one is equal to four. So question mark is equal to plus minus two. Know that sign of X is greater than general services equal to two. OK, so now which usar double ankle formula to reduce our tangent to X into two terrigino by over one minus tangents. Square to back. So what is tangents of X? What? That's equal to two over. What would you just So we had two times 2/1 minus two squared that I took with the 4/1 Mike, it's born which is equal to four over negative three. Okay, so this is our solution

So in this problem were asked to verify the given given equation. And so what were given here is were given Here is an equation and we're told to verify in other words, we want we want both sides to equal the exact same thing. So what? I mean by the exact same thing? Well, instead of for example, if I had 10 X is equal to sign X over coastline X. Well, I want to convert this such that it equals 2 10 x is equal to 10 x and this and this scenario in this area we want to get tangent of two x is equal to change it. Go to X injured of two X and so and so Let's look at our numerator first, while our numerator for our numerator we have to sign X coast on X And so remember that Remember that for our double angle formula This if we had we had to sign of Alfa Times Co sign of Alfa. This would give us this would give us sign of two Alfa. This would give a sign of two Alfa and so to sign Ex to sign ex coast on X could be rewritten as as sign of two X. And so now let's look at our denominator. So our denominator is is co sine squared X minus sine squared x So this looks very similar to A This looks very similar to our double angle identity for co sign of to two X So remember if we had we had co sign of two Alfa. This could equal to sequel to co sign squared Alfa. Coastlines were doubtful, minus sine squared off for And so we have this coastline of tooth data in Earth. So this gives us coast on two X in the denominator. And so putting this together we have we have tangent of two x is equal to is equal to sign of two x sign of two x over coastline two x and so will this problem becomes easier because remember, that sign over coastline is equal to attention. So this this sign is equal to change it two x And so we have verified that that tangent of two X is in fact equal to tangent of two x. So we have verified we have verified e equation

Mhm. Okay. Now I know the double angle formula for sign right. Sign two weeks is equal to to sign X. Because X. And I also know something about sine function and it is That sign is an odd function which means sign off minus X is nothing but minus sine X. This is an odd function, right? Why? Because Sign is an odd function. All right. So these are the properties that we are going to use. First of all. I have signed off -6. I have signed off -1. This is my left hand side by using this property of the heart function. This becomes minus cynics. All right. And using the double angle formula I know signed to accessible to Sinus Cossacks. So let me say that I'll put X instead of doing so what will if I put X instead of two X instead of X? I'll have to put X by two. Right, So this is going to be minus to sign X by two. Course X x two. And this actually is my right hand side. Okay, my right hand side hence proved. Mhm. Mhm. Mhm.


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