Question

cos(120+A)+cos(120-A) =-cosA

Answer 1

Hey there!

Let us solve this individually (Left hand side) to prove these trigonometric identities that is, L.H.S = R.H.S.

By applying the following trigonometric identity for "Cos(120 - A)" :

Here, Cos(s - t) = Cos(s) Cos(t) + Sin(s) Sin(t).

Cos(120 + A) + Cos(120) Cos(A) + Sin(120) Sin(A)

Apply the following trigonometric identities and trivial identities for "Sin(120)" and "Cos(120)" respectively that is,

Cos(x) = Sin(90 - x) and Sin(x) = Cos(90 - x)

Now,

Cos(120 + A) + Sin(90 - 130) Cos(A) + Cos(90 - 120) Sin(A)

$\mathsf{\bf{Cos(120 + A) - \frac{1}{2} Cos(A) + \frac{\sqrt{3}}{2} Sin(A)}}$

Again, apply the following trigonometric identity that is,

Here, Cos(s + t) = Cos(s) Cos(t) - Sin(s) Sin(t).

$\mathsf{\bf{Cos(120) Cos(A) - Sin(120) Sin(A) - \frac{1}{2} Cos(A) + \frac{\sqrt{3}}{2} Sin(A)}}$

Use this identity to obtain the value for "Cos(120)" :

Cox(x) = Sin(90 - x) and Sin(x) = Cos(90 - x)

Therefore,

$\mathsf{\bf{- \frac{1}{2} Cos(A) - \frac{\sqrt{3}}{2} Sin(A) - \frac{1}{2} Cos(A) + \frac{\sqrt{3}}{2} Sin(A)}}$

Add the similar elements and trigonometric values and identities to obtain the final value similar to R.H.S. :

$\mathsf{\bf{- 1 \times Cos(A) - \frac{\sqrt{3}}{2} Sin(A) + \frac{\sqrt{3}}{2} Sin(A)}}$

Add similar elements to prove the sides:

$\mathsf{\bf{- Cos(A)}}$

$\mathsf{\huge{\boxed{HENCE \: PROVED}}}}$

Hope this detailed answer helps you!

Answer 2

Answer:

cos(120+a)+cos(120-a)=-cosa