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2 sin 67,5 cos 67,5 = sin (2.67,5)
= sin 135
= sin (180-135)
= sin 45
= 1/2 √2
b) 1 - 2 sin²a = cos 2a
1 - 2 sin²15 = cos (2.15)
= cos 30
= 1/2 √3
c) sin² a - sin² b - cos² a + cos² b = (sin² a - cos² a) - ( sin² b - cos² b)
= - ( cos² a - sin ²a) + (cos² b - sin² b)
= - cos 2a + cos 2b
sin² 112,5 - sin² 22,5 - cos² 112,5 + cos² 22,5 = - cos (2.112,5) + cos (2. 22,5)
= - cos 225 + cos 45
= - ( - cos 45) + cos 45
= cos 45 + cos 45
= 2 cos 45
= 2. 1/2 √2
= √2
d) 2 tan a / 1 - tan ²a = tan 2a
2 tan 75 / 1- tan²75 = tan (2.75)
= tan 150
= - tan (180-150)
= - tan 30
= - 1/3 √3
e) cos 420 = cos (k.360 + α)
= cos (360 + 60)
= cos 60
√ [ (1- cos 420) / 2 ] = √ [ ( 1 - cos 60) / 2 ]
= √ [ ( 1 - 1/2) / 2 ]
= √ 1/4
= 1/2
f) cos 540 = cos (k.360 + α)
= cos (360 + 180)
= cos 180
√ [ (1+ cos 540) / 2 ] = √ [ (1 + cos 180) / 2 ]
= √ [ ( 1 - 1 ) / 2 ]
= 0
g) √ [ (1-cos 30) / (1+ cos 30) ] = √ [ ( 1 - 1/2) / (1+1/2) ]
= √ [ (1/2) / (3/2) ]
= √ 1/3
= 1/3 √3