Simplify each of the following 35 2 sin 15 cos 15 37 1 2 sin

Simplify each of the following. 35. 2 sin 15° cos 15° 37. 1 2 sin2 75° 39. sincos 1212 tan 22.5 1-tan 22.5° 41.

Solution

35. 2sin 15 cos 15

This is of the form

2 sinA cos A which is equal to sin2A

SO we get sin (2*15)=sin 30=1/2

37. 1-2sin²75

This is of the form 1-2sin²A=cos(2A)

so we get cos(2*75)=cos 150

-sqrt3/2

39. sin pi/2 cos pi/2

This is of the form sin A cos A=sin(2A)/2

So we get(1/2) sin(2*pi/2)=1/2*0=0

41. tan 22.5/(1-tan²22.5)

Which is of the form tan A/(1-tan²A) that is equal to tan (2A)

So we get

tan(2*22.5)=tan 45=1