Question

Prove that

Sin^2x + Cos^2x + 1 = 2Cos^2x + 2Sin^2x

Answer 1

you know that sin^2(x) + cos^2(x) is equal to one. So if you add another constant +1 to it what should turn out?

Answer 2

2

Answer 3

exactly, and the right hand side is just equal to two, when you factor the two out you're left with

2((cos^2(x)+sin^2(x))=2

Answer 4

but we are suppose to get sin^2(x) + cos^2(x) + 1 on other side too

Answer 5

You're supposed to verify that both sides are equal, on the left hand side you have a total of two, on the right hand side you have a total of 2, hence the two sides are equal.

Answer 6

Change the left side to 1 + 1 = 2, this is because sin^2(x) + cos^2(x) = 1. So you can replace that part with 1 and make the left side 2. Now we had a left side of 2 and right side of 2(cos^2(x) + sin^2(x)) and we must prove that they equal each other. So no we can say:\[R.S =2(\sin^2(x)+\cos^2(x))=2(1)=2=L.S\] And that works out to be just fine. The thing I did here basically was to show you how you don't have to keep a certain side of the identity exactly the way it's given; instead you can change it to an equivalent expression and then start proving the identity from there to make it easier for yourself. Here we changed the left to 2 so we could more easily prove the right side equal to the left side.

@pradiv

Answer 7

Do you understand?

Answer 8

kindoff thank you

Answer 9

No problem.

Answer 10

The point I am trying to get across here is: If you are having trouble proving an identity, change a side to an expression that you can easily prove the other side as. For example, if I am trying to prove the left side the same as right side but having problems, I change the left side to something that I can also change the right side to more easily. That way it becomes easier to prove the left and right sides, hence the identity.

Answer 11

if you are not busy can you solve from first on the Drawing Board by step so i will be able to other question by my self ? if it's okay with you

Answer 12

\[\sin^2(x) + \cos^2(x)+1=2(\sin^2(x) + \cos^2(x))\] I notice that it might be sort of annoying to prove the left and right sides using the current equation, perhaps if I changed the left hand side or right hand side in to something which I could also make the right hand side equal? That would make this easier to prove. So I decide to change my left hand side to 2. Because we know sin^2(x) + cos^2(x) = 1, then that part of the left side change be changed with 1 so it becomes 1 + 1 = 2. Now we are proving the identity:\[2 = 2(\sin^2(x) + \cos^2(x)) \]Notice how this is the same equation as the one we were given, it just looks different since we changed the left side so that we can more easily prove the identity. Now let's prove this. We will choose the right-side and prove it or show that it's the same as the left side:\[Right \ side = 2(\sin^2(x) + \cos^2(x))=2 (1)=2=Left \ side\] And this way we were able to show that in fact, the right side is equal to 2, which is the left side, and so the identity is proved.

@pradiv

Answer 13

@pradiv we have many ways to prove, you can go LHS ---> RHS
or RHS---> LHS
or LHS = something = RHS
the instruction above give you the 3rd method

Answer 14

but our teacher solve this way

Answer 15

can you show me how to solve that way ?

Answer 16

hey @ pradiv,, your note is another problem.

Answer 17

YOU drive the helpers crazy!!! look at the original one, it is different from your note. OMG

Answer 18

ok, it' s too late, but I can help you until you understand what happen.
ignore what you post, post the new one. Make sure you know what we are working with, don't mess the problems up, don't mix them ok? close this post and open another one. I'll be there. hurry, it's too late, your brother want to sleep.