Answer:
a) [tex]tan (157.5) = \frac{1-cos 315}{sin315}[/tex]
b)
[tex]sin (165) =\sqrt{ \frac{1-cos (330) }{2}}[/tex]
c)
[tex]sin^{2} (157.5) = \frac{1-cos (315) }{2}[/tex]
d)
cos 330° = 1- 2 sin² (165°)
Step-by-step explanation:
Step(i):-
By using trigonometry formulas
a)
cos2∝ = 2 cos² ∝-1
cos∝ = 2 cos² ∝/2 -1
1+ cos∝ = 2 cos² ∝/2
[tex]cos^{2} (\frac{\alpha }{2}) = \frac{1+cos\alpha }{2}[/tex]
b)
cos2∝ = 1- 2 sin² ∝
cos∝ = 1- 2 sin² ∝/2
[tex]sin^{2} (\frac{\alpha }{2}) = \frac{1-cos\alpha }{2}[/tex]
Step(i):-
Given
[tex]tan\alpha = \frac{sin\alpha }{cos\alpha }[/tex]
we know that trigonometry formulas
[tex]sin\alpha = 2sin(\frac{\alpha }{2} )cos(\frac{\alpha }{2} )[/tex]
1- cos∝ = 2 sin² ∝/2
Given
[tex]tan(\frac{\alpha }{2} ) = \frac{sin(\frac{\alpha }{2} )}{cos(\frac{\alpha }{2}) }[/tex]
put ∝ = 315
[tex]tan(\frac{315}{2} ) = \frac{sin(\frac{315 }{2} )}{cos(\frac{315 }{2}) }[/tex]
multiply with ' 2 sin (∝/2) both numerator and denominator
[tex]tan (\frac{315}{2} )= \frac{2sin^{2}(\frac{315)}{2} }{2sin(\frac{315}{2} cos(\frac{315}{2}) }[/tex]
Apply formulas
[tex]sin\alpha = 2sin(\frac{\alpha }{2} )cos(\frac{\alpha }{2} )[/tex]
1- cos∝ = 2 sin² ∝/2
now we get
[tex]tan (157.5) = \frac{1-cos 315}{sin315}[/tex]
b)
[tex]sin^{2} (\frac{\alpha }{2}) = \frac{1-cos\alpha }{2}[/tex]
put ∝ = 330° above formula
[tex]sin^{2} (\frac{330 }{2}) = \frac{1-cos (330) }{2}[/tex]
[tex]sin^{2} (165) = \frac{1-cos (330) }{2}[/tex]
[tex]sin (165) =\sqrt{ \frac{1-cos (330) }{2}}[/tex]
c )
[tex]sin^{2} (\frac{\alpha }{2}) = \frac{1-cos\alpha }{2}[/tex]
put ∝ = 315° above formula
[tex]sin^{2} (\frac{315 }{2}) = \frac{1-cos (315) }{2}[/tex]
[tex]sin^{2} (157.5) = \frac{1-cos (315) }{2}[/tex]
d)
cos∝ = 1- 2 sin² ∝/2
put ∝ = 330°
[tex]cos 330 = 1 - 2sin^{2} (\frac{330}{2} )[/tex]
cos 330° = 1- 2 sin² (165°)
