Homework proving and solving identities

K

Kellyanne Smith

New member
Joined
Nov 23, 2021
Messages
4
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)
 
Last edited:
Kellyanne Smith said:
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)
Click to expand...
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.
 
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
 
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?
 
Kellyanne Smith said:
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)
Click to expand...
..and the x (in 45-x) is also in degrees. Best to say that the whole angle, 45-x, is in degrees.
 
Kellyanne Smith said:
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
Click to expand...
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.....
 
Jomo said:
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?
Click to expand...

Subhotosh Khan said:
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.....
Click to expand...
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?
 
Kellyanne Smith said:
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?
Click to expand...
No! - Look into that step.
 
You must log in or register to reply here.
Top