Homework proving and solving identities

Kellyanne Smith

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Nov 23, 2021
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Hello I need help with the following question I have an idea on how to go on but for some reason, I am having issues making them the same
1)Prove that sin 2x + 2 sin^2(45 - x ) = sin^2 x + cos^2 x
(the 45 is in degrees)
 
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Dr.Peterson

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Nov 12, 2017
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Hello I need help with the following question I have an idea on how to go on but for some reason, I am having issues making them the same
1)Prove that sin 2x + 2 sin^2(45 - x ) = sin^2 x + cos^2 x
(the 45 is in degrees)
Please show us what you've done, so we can nudge you in the right direction or correct errors. We need to see your "issues" in order to deal with them.

I assume you've used the angle-subtraction identity and the double-angle identity.
 

Kellyanne Smith

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yes i have i replaced the sin 2x with 2 sinx cos x and replaced the sin^2(45-x) with (sqrt 2/2 cos x - sqrt 2/2 sin x) all squared.
then I went along simplifying the expression I have rewritten the (sqrt 2/2 cos x - sqrt 2/2 sin x) all squared as a whole fraction (sqrt 2 cos x - sqrt 2 sin x / 2) then I raised what is in the bracket to the power of 2. and from there I began to doubt what I was doing
 

Jomo

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You stopped just where you claim you are having trouble! Please show us your work so we can find your error. Do you know what the right hand side reduces to?
 

Jomo

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Hello I need help with the following question I have an idea on how to go on but for some reason, I am having issues making them the same
1)Prove that sin 2x + 2 sin^2(45 - x ) = sin^2 x + cos^2 x
(the 45 is in degrees)
..and the x (in 45-x) is also in degrees. Best to say that the whole angle, 45-x, is in degrees.
 

Subhotosh Khan

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yes i have i replaced the sin 2x with 2 sinx cos x and replaced the sin^2(45-x) with (sqrt 2/2 cos x - sqrt 2/2 sin x) all squared.
then I went along simplifying the expression I have rewritten the (sqrt 2/2 cos x - sqrt 2/2 sin x) all squared as a whole fraction (sqrt 2 cos x - sqrt 2 sin x / 2) then I raised what is in the bracket to the power of 2. and from there I began to doubt what I was doing
so you did the following:

sin 2x + 2 sin^2(45 - x )

= 2 * sin(x) * cos(x) + 2 * \(\displaystyle \left[\frac{\sqrt{2}}{2} * cos(x) - \frac{\sqrt{2}}{2} * sin(x) \right]^2 \)

After that I, I would factor out the (√2)/2 and realize that [(√2)/2]2 = 1/2 and continue.....
 

Kellyanne Smith

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Nov 23, 2021
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You stopped just where you claim you are having trouble! Please show us your work so we can find your error. Do you know what the right hand side reduces to?

so you did the following:

sin 2x + 2 sin^2(45 - x )

= 2 * sin(x) * cos(x) + 2 * \(\displaystyle \left[\frac{\sqrt{2}}{2} * cos(x) - \frac{\sqrt{2}}{2} * sin(x) \right]^2 \)

After that I, I would factor out the (√2)/2 and realize that [(√2)/2]2 = 1/2 and continue.....
I have continued and now after factoring out the (√2)/2 and realize that [(√2)/2]2 = 1/2
I have 2 sinx cos x + 2 (1/2 cos^2x - 2sinxcosx + 1/2sin^2x)
reduced to 2sinx cos x + cos^2 x - 2sinxcosx + sin^2x
= sin^2x + cos^2x

is this correct?
 

Subhotosh Khan

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I have continued and now after factoring out the (√2)/2 and realize that [(√2)/2]2 = 1/2
I have 2 sinx cos x + 2 (1/2 cos^2x - 2sinxcosx + 1/2sin^2x)
reduced to 2sinx cos x + cos^2 x - 2sinxcosx + sin^2x
= sin^2x + cos^2x

is this correct?
No! - Look into that step.
 
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