Use the producttosum identities to rewrite the following exp

Use the product-to-sum identities to rewrite the following expression.

cos26°cos43°

Simplify your answer. Do not evaluate.

Solution

given

cos26°cos43°

we know

As cos(A+B)= cosAcosBsinAsinB and

cos(AB)=cosAcosB+sinAsinB

adding the two we get cos(A+B)+cos(AB)=2cosAcosB

hence

==> ( 1/2) 2 cos26°cos43°

==> ( 1/2) [ cos( 26° + 43° )+cos( 26° - 43° ) ]

==>( 1/2) [ cos( 69° )+cos( - 17° ) ] ( since cos ( - x ) = cos x )

==> ( 1/2) [ cos( 69° ) + cos( 17° ) ]

Hence cos26°cos43° = ( 1/2) [ cos( 69° ) + cos( 17° ) ]