Question

prove geometically that cos(x+y) =cosxcosy-sin xsin y and hence find cos (π/2+x)​

Answer 1

Answer:

cosπ/2cosx-sinπ/2sinx

0-sinx

-sinx

Answer 2

Answer:

What is the proof that cos (x + y) = cos (x) cos (y) − sin (x) sin (y)?

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.

Construction: On the bounding line of the compound angle (α + β) take a point A on OZ, and draw AB and AC perpendiculars to OX and OY respectively. Again, from C draw perpendiculars CD and CE upon OX and AB respectively.

Proof: From triangle ACE we get, ∠EAC = 90° - ∠ACE = ∠ECO = alternate ∠COX = α.

Now, from the right-angled triangle AOB we get,

cos (α + β) = OB/OA

= OD−BD/OA

= OD/OA - BD/OA

= OD/OA - EC/OA

= ( OD/OC ∙ OC/OA) - ( EC/AC ∙ AC/OA)

= cos α cos β - sin ∠EAC sin β

= cos α cos β - sin α sin β, (since we know, ∠EAC = α)

Therefore, cos (α + β) = cos α cos β - sin α sin β. Proved

Well, there is another good proof given in NCERT..(which may be a bit simpler)..So, you may refer to that for more transparency of the proof..

Hope this was helpful

Regards.

Step-by-step explanation: