Question

If `sin alpha + sin beta + sin gamma = cos alpha + cos beta + cos gamma` then what is the value of `cos^2 alpha + cos^2 beta + cos^2 gamma` ?

Answer 1

There is insufficient information to determine a single value.

Explanation:

The question gives insufficient information to determine a unique value.

For example:

If `(alpha, beta, gamma) = (0, pi, pi/4)` then:

`{ (sin alpha + sin beta + sin gamma = 0+0+sqrt(2)/2 = sqrt(2)/2), (cos alpha + cos beta + cos gamma = 1-1+sqrt(2)/2 = sqrt(2)/2), (cos^2 alpha+cos^2beta+cos^2gamma = 1+1+1/2 = 5/2) :}`

If `(alpha, beta, gamma) = (pi/4, pi/4, pi/4)` then:

`{ (sin alpha + sin beta + sin gamma = sqrt(2)/2+sqrt(2)/2+sqrt(2)/2 = (3sqrt(2))/2), (cos alpha + cos beta + cos gamma = sqrt(2)/2+sqrt(2)/2+sqrt(2)/2 = (3sqrt(2))/2), (cos^2 alpha+cos^2 beta + cos^2 gamma = 1/2+1/2+1/2 = 3/2) :}`

Are `alpha, beta, gamma` supposed to be the internal angles of a triangle or something like that? Is there some information missing from the question?

Answer 2

Subject to `alpha +beta +gamma =pi`\
`cos^2 alpha +cos^2 beta +cos^2 gamma= 3/2 +cos alpha +cos beta +cos gamma = 3/2 +sin alpha +sin beta +sin gamma`

Explanation:

also `cos^2 alpha +cos^2 beta+ cos^3 gamma = 3/2 + sin alpha +sin beta + sin gamma`