Cos(x+y+z) Formula, Proof | Cos(x+y+z) Identity

The cos(x+y+z) formula is given by cos(x+y+z) = cosx cosy cosz – sinx siny cosz – sinx cosy sinz – cosx siny sinz. Here we establish the identity of cos(x+y+z).

Formula of cos(x+y+z)

The formula of cos(x+y+z) is given below:

$\cos(x+y+z) = \cos x \cos y \cos z – \sin x \sin y \cos z – \sin x \cos y \sin z – \cos x \sin y \sin z$.

Proof:

In order to obtain the formula of cos(x+y+z), we will use the formulas:

1. cos(a+b) = cosa cosb – sina sinb
2. sin(a+b) = sina cosb + cosa sinb

Now, cos(x+y+z)

= cos ((x+y)+z)

= cos(x+y) cosz – sin(x+y) sinz, obtained by using the above formula 1 with a=x+y and b=z.

= [cosx cosy – sinx siny] cosz – [sinx cosy + cosx siny] sinz, by the above two formuals.

= cosx cosy cosz – sinx siny cosz – sinx cosy sinz – cosx siny sinz.

So the formula of cos(x+y+z) formula is equal to cos(x+y+z) = cosx cosy cosz – sinx siny cosz – sinx cosy sinz – cosx siny sinz, which is obtained by applying the formulas of cos(a+b) and sin(a+b).

More Trigonometric Formulas: Sin(x+y+z) Formula, Proof

cos(a+b) cos(a-b) Formula

sin(a+b) sin(a-b) Formula

cosx cosy Formula	sinx siny Formula
cosx siny Formula	sinx cosy Formula
cosec2x Formula	sec2x Formula

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