need help with proving this trig "identity"

A

abel muroi

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I was given this,

(sin x + cos x)2 = 1 + 2 sin x cos x

and was told to prove it.

i know the left side of this equation is the same as one but i am not entirely sure about the right side.

what am i supposed to do with the 2? what happens if i multiply sin x and cos x? is that even legal?
 
abel muroi said:
I was given this,

(sin x + cos x)2 = 1 + 2 sin x cos x

and was told to prove it.

i know the left side of this equation is the same as one but i am not entirely sure about the right side.

what am i supposed to do with the 2? what happens if i multiply sin x and cos x? is that even legal?
Click to expand...

What is (sin x + cos x)2, it is NOT one.
 
Ishuda said:
So what would that be if you multiplied it out?
Click to expand...

thats actually where im a little confused... if i multiply sin x and cos x, im not sure what the result will be

btw, if i multiply sin x and sin x... will i get sin2 x or 2sin x?
 
abel muroi said:
thats actually where im a little confused... if i multiply sin x and cos x, im not sure what the result will be

btw, if i multiply sin x and sin x... will i get sin2 x or 2sin x?
Click to expand...

Just as you multiply out
(a+b) (a+b) = a(a+b) + b(a+b)
= aa + ab + ba + bb
= a2 + 2 ab + b2,
you will get sin2(x) and the same with the cosine.
 
wait i think i have figured it out..

(sin x + cos x) (sin x + cos x)

sin2 x + sin x cos x + sin x cos x + cos2 x

(then i combine cos2 x and sin2 x, and i add sin x cos x + sin x cos x)

1 + 2 sin x cos x


is this the correct way to prove this trig equation?
 
abel muroi said:
wait i think i have figured it out..

(sin x + cos x) (sin x + cos x)

sin2 x + sin x cos x + sin x cos x + cos2 x

(then i combine cos2 x and sin2 x, and i add sin x cos x + sin x cos x)

1 + 2 sin x cos x


is this the correct way to prove this trig equation?
Click to expand...
Yes, very good!
 
abel muroi said:
i have one more question...

if i am asked to prove a trig equation, do you think its always best to start from the left side? or does it matter?
Click to expand...
Not always. It depends on the problem. Sometimes you have to start on both sides and meet in the middle.
 
Ishuda said:
Not always. It depends on the problem. Sometimes you have to start on both sides and meet in the middle.
Click to expand...

i have another question

if i subtract 1 and cos2 x, what will the result be?
 
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abel muroi said:
i have another question

if i subtract 1 and cos2 x, what will the result be?
Click to expand...

If you mean "1 - cos2(x)" - then use very well known trig-identity

sin2(x) + cos2(x) = 1

and get your solution....
 
abel muroi said:
so 1 - cos2 (x) = sin2 (x)

but i dont understand WHY that equals sin2 (x)

can you explain that to me?
Click to expand...

If A + B = 1 → B = 1-A ... and that's it.

Are you asking - why sin2(x) + cos2(x) = 1 ?
 
abel muroi said:
i am asking why 1 - cos x = sin x.. ............ It is NOT - where did you see that

so the same would apply if i did this..

1 - sin x = cos x

and

1 + cos x = sinx

right?
Click to expand...

1 - cos(x) * cos(x) = sin(x) * sin(x)
 
abel muroi said:
what about 1 + cos x?

you said that doesn't equal to sin x, so what does it equal?
Click to expand...
1 + cosx can not be simplified.
I think that you might be using the wrong rule: if a + b = c then sqrta + sqrt b = sqrt c. NOT true.
1 - sin^2(x) = cos^2(x) then sqrt1 - sqrt(sin^2(x)) = sqrt(cos^2(x)) then 1-sinx=cosx. The 1st then is not valid.
As a student whenever I thought that (or something similar to that) was true I would think then why would the Pythagorean be a^2 + b^2 = c^2 when this is the same as a+b=c. Answer: because it is NOT true!
 
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