Proving sin4x + 2sin2x = 8sinxcos^3x Using Trigonometric Identities

[Attis]

Homework Statement

show that sin4x + 2sin2x = 8sinxcos^3x

Homework Equations

sin2x =2sinxcosx

The Attempt at a Solution

I started out by letting sin4x = sin(2x*2) so that I could plugg in sin2x = 2sinxcosx in the equation.
sin(2x*2) +2sin2x =
sin2*sin2x +2sin2x =
sin2 * (2sinxcosx) + 2*sin2x =
sin2* (2sinxcosx) + 2* (2sinxcosx)
and I end up with 8sinxcosx
which is wrong!
??

 

[Tanya Sharma]

sin(2x*2) +2sin2x =
sin2*sin2x +2sin2x =
sin2 * (2sinxcosx) + 2*sin2x =
sin2* (2sinxcosx) + 2* (2sinxcosx)
and I end up with 8sinxcosx
which is wrong!
??

sin(ab) ≠ (sina)(sinb) i.e sin4x≠(sin2)(sin2x)

sin4x = 2sin2xcos2x

 

[Attis]

sin(ab) ≠ (sina)(sinb) i.e sin4x≠(sin2)(sin2x)

sin4x = 2sin2xcos2x

Ok, did you derive that from sin2x = 2sinxcosx?
In that case, I don´t understand why sin4x = 2sin2xcos2x and not 4sin2xcos2x (i.e twice as much as sin2x = 2sinxcosx?)

 

[Tanya Sharma]

Ok, did you derive that from sin2x = 2sinxcosx?

Yes

In that case, I don´t understand why sin4x = 2sin2xcos2x and not 4sin2xcos2x (i.e twice as much as sin2x = 2sinxcosx?)

Please answer this - What is sin2A ?

 

[Attis]

Yes



Please answer this - What is sin2A ?

2sinAcosA?

Or is this some sort of a trick question?

 

[Tanya Sharma]

Or is this some sort of a trick question?

Not at all . Just trying to make you understand the formula

2sinAcosA?

sin2A = 2sinAcosA . Now replace A with 2x on both the sides .What do you get ?

 

[Attis]

Not at all . Just trying to make you understand the formula



sin2A = 2sinAcosA . Now replace A with 2x on both the sides .What do you get ?

Ah! I see. Thanks!
sin2*2x= 2sin2xcos2x.
I´ll carry on now and see if I get anywhere.

 

[Attis]

Somehow it keeps on getting more and more complicated. Do you have any idea on how I could keep it "simple"?

sin4x + 2sin2x = 8sinxcos^3x
on the left hand side:
2sin2xcos2x + 2sin2x =
2*2sinxcosx(cos^2x - sin^2x) + 2*2sinxcosx =
4sinxcos^3x - 2sin^3xcosx + 4sinxcosx

It really doesn´t feel right...

 

[Tanya Sharma]

2sin2xcos2x + 2sin2x

Is there something common in the two terms ? If yes ,take out the common factor .What do you get ?

 

[Attis]

Is there something common in the two terms ? If yes ,take out the common factor .What do you get ?

2sin2x(cos2x + 1)?

 

[Tanya Sharma]

Good...

1+cos2x = ?

 

[Attis]

1 + cos^2x - sin^2x
?

 

[Tanya Sharma]

1 + cos^2x - sin^2x
?

Correct...Is there some other way of expressing this expression ? Can a couple of terms be combined ?

 

[Attis]

There are two options:
1) 1 + cos^2x -sin^2x =
1+cos^2x - (1-cos^2x)=2cos^2x

2) 1+(1-sin^2x)-sin^2x=
1+1 - sin^2x -sin^2x = 2-2sin^2x = 2(1-sin^2x)

?

 

[Tanya Sharma]

There are two options:
1) 1 + cos^2x -sin^2x =
1+cos^2x - (1-cos^2x)=2cos^2x

Right...

Now put sin2x = 2sinxcosx and (1+cos2x) = 2cos²x in the expression you have in post#10.

 

[Attis]

Right...

Now put sin2x = 2sinxcosx and (1+cos2x) = 2cos²x in the expression you have in post#10.

Perfect! Now I got it.
Thanks a lot! you´ve been a massive help.
Do you have any general tips on how to solve such problems?

 

[Tanya Sharma]

Remembering trigonometric formulas and practicing variety of problems of increasing complexity is the key .If you remember the formulas then it will help you in manipulating the trigonometric expressions .

You should be comfortable with double ,triple angle formulas .How to convert from one form to other .

Cos2x = cos²x-sin²x = 1-2sin²x = 2cos²x-1 .

sinx = √[(1-cos2x)/2] ; cosx = √[(1+cos2x)/2]

All the above are various ways of writing the same thing .With practice you will recognize which form to use .

 

[Attis]

Remembering trigonometric formulas and practicing variety of problems of increasing complexity is the key .If you remember the formulas then it will help you in manipulating the trigonometric expressions .

You should be comfortable with double ,triple angle formulas .How to convert from one form to other .

Cos2x = cos²x-sin²x = 1-2sin²x = 2cos²x-1 .

sinx = √[(1-cos2x)/2] ; cosx = √[(1+cos2x)/2]

All the above are various ways of writing the same thing .With practice you will recognize which form to use .

Ok. I´ll do my best. Thanks once again!