Trigonometry?

Question

Prove this identities:
1.sin(a+30)-sin(a-30)=cos a
2.cos(45+a)+cos(135+a)=-sqrt2*sin a

Answer 1

1.) sin(a+30)-sin(a-30)=cos a

sin (a+30) = sin(a) cos (30) + cos(a) sin(30)
sin (a - 30) = sin(a) cos (30) - cos(a) sin(30)
sin(30) = 0.5
cos(30) = sqrt(3)/2 = 0.866

sin(a+30)-sin(a-30) = sin(a)(sqrt(3)/2) + cos(a)(0.5) - sin(a)(sqrt(3)/2) + cos(a)(0.5) = cos(a) << answer.

2.) cos(45+a)+cos(135+a)

cos(45+a) = cos(45) cos(a) - sin(45)sin(a)
cos(135+a) = cos(135)cos(a) - sin(135)sin(a)
cos(45) = 1/sqrt(2)
sin(45) = 1/sqrt(2)
cos(135) = -1/sqrt(2)
sin(135) = 1/sqrt(2)

=cos(45) cos(a) - sin(45)sin(a) + cos(135)cos(a) - sin(135)sin(a)

= cos(a)/sqrt(2) - sin(a)/sqrt(2) - cos(a)/sqrt(2) - sin(a)/sqrt(2)
= -2sin(a)/sqrt(2)
= -2sin(a)(sqrt (2))/2
= -sin(a)sqrt(2) << answer

Answer 2

sinacos30+cosasin30 -(sinacos30-cosasin30)=
2cosasin30 = (2cosa) (.5) =cosa

cos(45+a)+cos(135+a)=-sqrt2*sina
LHS = cos45cosa-sin45sina+cos135cosa-sin135sina
LHS =rt(2)/2cosa - rt(2)/2sina -rt(2)/2cosa -rt(2)/2sina
LHS = -2rt(2)/2sina = -rt(2)sina = RHS