# Why is sin (x+x) = sinx cosx + cosx sinx ?

1. Mar 4, 2006

### Natasha1

Why is sin (x+x) = sinx cosx + cosx sinx ? Simple explanation required please

2. Mar 4, 2006

### rindech

sin (x+y) = sin x cos y + cos x sin y. Simply place an "x" for the "y" in the formula. Noted: sin (2x) = 2sin x cos x.

3. Mar 4, 2006

### LeonhardEuler

I don't know if you are familiar with Euler's formula, but if you are then those trig formulas are easy to derive:
$$e^{ix}=\cos{x}+i*\sin{x}$$
so
$$e^{i2x}=(\cos{x}+i*\sin{x})^2$$
$$=\cos^2{x}+2i*\sin{x}*\cos{x}-\sin^2{x}$$
Since
$$\sin{x}=Im:e^{ix}$$
Then
$$\sin{2x}=2*\sin{x}\cos{x}$$
You also get the double angle formula for cosine for free. If you do not know Euler's formula, then you can still prove this geometrically, but it will take more work.

4. Mar 5, 2006

### VietDao29

???
This is not correct.
What if I say that: sin(x + y) = sin(x)sin(y) + cos(x)cos(y) + sin(x)cos(y) + sin(y)cos(x) - 1.
It certainly satisfies: sin(2x) = sin(x + x) = 2sin(x)cos(x). But it's not true, right?
And moreover, it's some kind of circular argument. One should know the angle sum identities before they know the double identities.
There's a geometry proof at the end of this article. (it works for 0 <= x, y <= 90o). One can then show that the identity is true for every angle.

Last edited: Mar 5, 2006
5. Mar 5, 2006

### d_leet

It's certainly true if x=y but not in general. But sin(2x) does equal 2sin(x)cos(x) because sin(x+y) = sin(x)cos(y) + cos(x)siny(y)