Question

cos ( x+y) derivation​

Answer 1

HERE IS YOUR ANSWER :

The expansion of cos (α + β) is generally called addition formulae. In the geometrical proof of the addition formulae we are assuming that α, β and (α + β) are positive acute angles. But these formulae are true for any positive or negative values of α and β.

Now we will prove that, cos (α + β) = cos α cos β - sin α sin β; where α and β are positive acute angles and α + β < 90°.

Let a rotating line OX rotate about O in the anti-clockwise direction. From starting position to its initial position OX makes out an acute ∠XOY = α.

Again, the rotating line rotates further in the same direction and starting from the position OY makes out an acute ∠YOZ = β.

Thus, ∠XOZ = α + β < 90°.

We are suppose to prove that, cos (α + β) = cos α cos β - sin α sin β.