geometrically prove that :

1.sin(X-Y) = sinx.cosy-cosx-siny

2. sin(x+y)=sinx.cosy + cosx.siny

3, cos(x+y)= cosx.cosy-sinx.siny

4.tan(x+y)= tanx+tany/1-tanxtany

5. tan(x'y) =tanx-tany/1+tanxtany

Here is the answer for two of the proofs.

2. sin(x+y)=sinx.cosy + cosx.siny

3, cos(x+y)= cosx.cosy-sinx.siny


Let OX be the initial line.
Let So, Let us choose a point P on OZ.
Draw PM perpendicular to OX and PQ perpendicular to OY. From the point O, draw QN perpendicular to OX and QR perpendicular to PM.

now,
From triangle MOP,
iSinA+B=PMOP=MR+RPOP=MROP+RPOP=QNOP+RPOP=QNOQ×OQOP+RPPQ×PQOP=sinA. Cos B +Cos A . sinBii cos A+B=OMOP=ON-RQOP=ONOP-RQOP=ONOQ×OQOP-RQPQ×PQOP=cosA. CosB - sinA sin B

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