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The expression cos^(2)(alpha + beta + ga...

The expression `cos^(2)(alpha + beta + gamma) + cos^(2)(beta + gamma) + cos^(2)alpha - 2cos alpha cos(beta + gamma)cos(alpha + beta + gamma)` is

A

independent of `alpha`

B

independent of `beta`

C

dependent on `gamma` only

D

dependent on `alpha, beta, gamma`

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The correct Answer is:
A, B
`E = cos^(2)alpha + cos^(2)(beta + gamma) + cos^(2)(alpha + beta + gamma) - 2cos alpha cos(beta + gamma)cos(alpha + beta + gamma)`
`= cos^(2)alpha + cos^(2)(beta + gamma) + cos^(2)(alpha + beta + gamma) - [(cos(alpha + beta + gamma) + cos(alpha - beta - gamma)).cos(alpha + beta + gamma)]`
`= cos^(2)alpha + cos^(2)(beta + gamma) - [cos(beta + gamma - alpha) cos (beta + gamma + alpha))`
`= cos^(2)alpha + cos^(2)(beta + gamma) - (cos^(2)(beta + gamma) - [cos^(2)(beta + gamma) - sin^(2)alpha ] = 1`
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