Cos (x+y) =

Answers
Answer:
cos(x + y) = cosxcosy - sinxsiny.
Step-by-step explanation:
Proof of cos( x + y ) = cosxcosy - sinxsiny : Using sin( A + B ) = sinAcosB + cosAsinB
x is written as A and y is written as B.
From the properties of trigonometry :
- sin( A + B ) = sinAcosB + cosAsinB
- 1 - sin^2 A = cos^2 A
= > cos( A + B )
= > √{ cos^2 ( A + B ) }
= > √{ 1 - sin^2 ( A + B ) } { cos^2 ( A + B ) = 1 - sin^2 ( A + B ) }
= > √{ 1 - ( sinAcosB + cosAsinB )^2 } { sin( A + B ) = sinAcosB + cosAsinB }
= > √[ 1 - ( sin^2 A cos^2 B + cos^2 A sin^2 B + 2sinAcosBsinBcosA ) ]
= > √[ sin^2 A + cos^2 A - ( sin^2 A cos^2 B + cos^2 A sin^2 B + 2sinAcosBsinBcosA ) ] { 1 = sin^2 A + cos^2 A }
= > √[ sin^2 A + cos^2 A - sin^2 A cos^2 B - cos^2 A sin^2 B - 2sinAcosBsinBcosA ]
= > √[ sin^2 A( 1 - cos^2 B ) + cos^2 A( 1 - sin^2 B ) - 2sinAsinBcosAcosB ]
= > √[ sin^2 A.sin^2 B + cos^2 A.cos^2 B
- 2cosAcosBsinAsinB ]
= > √[ ( cosAcosB - sinAsinB )^2 ] { cosine is taken as positive, as value of cosine decreases on increasing the angle }
= > cosAcosB - sinAsinB