Trig Identity proof

Show that:

$4(\sin^4x+\cos^4x) \equiv \cos4x + 3$.

Really stuck with this, no idea how to go ahead with it. The book gives a hint: $\sin ^4 x = (\sin ^2 x)^2$ and use $\cos 2x = 1 - 2\sin ^2 x$

But I don't even understand the hint, where did they get
$\cos 2x = 1 - 2\sin ^2 x$ from?

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Mark44
Mentor
Show that:

$4(\sin^4x+\cos^4x) \equiv \cos4x + 3$.

Really stuck with this, no idea how to go ahead with it. The book gives a hint: $\sin ^4 x = (\sin ^2 x)^2$ and use $\cos 2x = 1 - 2\sin ^2 x$

But I don't even understand the hint, where did they get
$\cos 2x = 1 - 2\sin ^2 x$ from?
This is one of three double-angle identities for cos(2x).

The other two are
cos(2x) = cos2(x) - sin2(x)
cos(2x) = 2cos2(x) - 1

The 2nd one above and the one you're asking about can be obtained from the first one I showed by using the identity sin2(x) + cos2(x) = 1.

CAF123
Gold Member
But I don't even understand the hint, where did they get
cos2x=1−2sin2x from?
$$cos (x + x) = cos^2x - sin^2x = (1 - sin^2x) -sin^2x = 1 - 2sin^2x,$$ using the identity $sin^2x +cos^2x =1$

CAF123
Gold Member
To start you off, write the given equation as, $$4((\sin^2x)^2 + (\cos^2x)^2),$$ and use the double angle identities to get this solely in terms of $\cos.$

You can start by using this:
a2+b2=(a+b)2-2ab
This would mean that:
$$4(\sin^4x+\cos^4x)=4((sin^2x+cos^2x)^2-2sin^2xcos^2x)$$
Now rest should be easy to solve.

To start you off, write the given equation as, $$4((\sin^2x)^2 + (\cos^2x)^2),$$ and use the double angle identities to get this solely in terms of $\cos.$
working :

$4((sin^2x)^2 + (cos^4x)$
$4((1-cos^2x)^2 + cos^4x))$
$4(1-2cos^2x + cos^4x + cos^4x)$
$4(1-2cos^2x + 2cos^4x)$

Pretty much stuck here.
You can start by using this:
a2+b2=(a+b)2-2ab
This would mean that:
$$4(\sin^4x+\cos^4x)=4((sin^2x+cos^2x)^2-2sin^2xcos^2x)$$
Now rest should be easy to solve.
continuing from your working: $4(1 - 2sin^2xcos^2x)$, but I don't really know any identities to go from here...

Bacle2
How about doing the angle sum on the RHS: Cos4x= Cos(2x+2x) , and expand?

How about doing the angle sum on the RHS: Cos4x= Cos(2x+2x) , and expand?
He doesn't have $cos(4x)$, he has $cos^{4}(x)$, or $(cos(x))^{4}$

ehild
Homework Helper
working :

$4((sin^2x)^2 + (cos^2x)^2)$
Use the following double-angle identities:
sin2(x)=0.5(1-cos(2x))
cos2(x)=0.5(1+cos(2x))

ehild

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Mark44
Mentor
How about doing the angle sum on the RHS: Cos4x= Cos(2x+2x) , and expand?
He doesn't have $cos(4x)$, he has $cos^{4}(x)$, or $(cos(x))^{4}$
Bacle2 explicitly was referring to the right side of the original identity, which has a cos(4x) term in it.

See below.

$4(\sin^4x+\cos^4x) \equiv \cos4x + 3$.

Bacle2
He doesn't have $cos(4x)$, he has $cos^{4}(x)$, or $(cos(x))^{4}$
Look at the RH side of the original post, it has a Cos(4x)+3.

Edit: Sorry, I did not see Mark44's post, and now I cannot see the 'delete' option.

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continuing from your working: $4(1 - 2sin^2xcos^2x)$, but I don't really know any identities to go from here...
Try using these identities:
i)sin(2x)=2sin(x)cos(x)
ii)cos(2x)=1-2sin2x

If you square both the sides of i identity, you may see a way through.

CAF123
Gold Member
Use what ehild suggested if attempting the way I started (which is just one way to tackle the problem - there are many others).
We have, as suggested in my last post,
$4((\sin^2x)^2 + (\cos^2x)^2)$

Now, use the double angle identities that ehild suggested, specifically $\sin^2x = -\frac{1}{2}(\cos2x - 1)$ and $\cos^2x = \frac{1}{2}(cos2x +1)$

Substitute these into the eqn, expand the brackets and simplify...