Question

Prove that
4 sin 15°sin 75º = √2 (cos 105° + sin 75°)​

Answer 1

Answer:

Step-by-step explanation:

cos105=cos(90+15)=-sin15

sin75=sin(90-15)=cos15

sin15=sin(45-30)=sin45cos30-cos45sin30

=(sqrt(3)/2sqrt(2))-(1/2sqrt(2))

=sqrt(3)-1/2sqrt(2)

sin75=sin(30+45)=sin30cos45+cos30sin45=(1/2sqrt(2))+(sqrt(3)/2sqrt(2))

=sqrt(3)+1/2sqrt(2)

LHS:

4sin15sin75=4*(sqrt(3)-1))(sqrt(3)+1)/8=3-2/2=1/2

RHS:

sqrt(2)((-(sqrt(3))+1/2sqrt(2))+(sqrt(3)+1/2sqrt(2))

=(sqrt(2)/2sqrt(2))

=1/2

LHS=RHS