To determine
(a)
To obtain:
Double angle formula for sine function.
Answer
sin2x=2sinxcosx
Explanation
1) Formula:
i. Sum rule:
ddxfx+gx=ddxfx+ddxg(x)
ii. Difference rule:
ddxfx-gx=ddxfx-ddxg(x)
iii. Power rule combined with chain rule:
ddxfxn=n.fxn-1.f'(x)
iv. ddxcosf(x)=-sinfx.f'(x)
v. ddxsinf(x)=cosfx.f'(x)
2) Given:
cos2x=cos2x-sin2x
3) Calculation:
cos2x=cos2x-sin2x
Differentiate given equation with respect to x,
ddxcos2x=ddx(cos2x-sin2x)
By using difference rule,
ddxcos2x=ddxcos2x-ddxsin2x
By using power rule combined with chain rule and using rule(iv),
-2sin2x=2.cosx-sinx-2.sinxcosx=-4.sinxcosx
sin2x=2sinxcosx
Conclusion:
Double angle formula for sine function: sin2x=2sinxcosx
To determine
(b)
To obtain:
Addition formula for cosine function.
Answer
cos(x+a)=cosa.cosx-sina.sinx
Explanation
1) Given:
sin(x+a)=sinxcosa+cosxsina
2) Calculation:
Differentiate given equation with respect to x,
ddxsin(x+a)=ddx[sinxcosa+cosxsina]
By using sum rule and using rule (iv),(v)
cosx+a.ddxx+a=ddxsinxcosa+ddxcosxsina
cos(x+a)*1=cosa.ddxsinx+sina.ddxcosx
cos(x+a)=cosa.cosx+sina.-sinx
cos(x+a)=cosa.cosx-sina.sinx
cos(x+a)=cosa.cosx-sina.sinx
Conclusion:
Addition formula for cosine function: cos(x+a)=cosa.cosx-sina.sinx