Question

Cos (x+y) =

$cosx cosy - sinx siny prove it$
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Answer 1

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