To determine

(a)

To obtain:

Double angle formula for sine function.

Answer

sin⁡2x=2sin⁡xcos⁡x

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. ddxcos⁡f(x)=-sin⁡fx.f'(x)

v. ddxsin⁡f(x)=cos⁡fx.f'(x)

2) Given:

cos⁡2x=cos2⁡x-sin2⁡x

3) Calculation:

cos⁡2x=cos2⁡x-sin2⁡x

Differentiate given equation with respect to x,

ddxcos⁡2x=ddx(cos2⁡x-sin2⁡x)

By using difference rule,

ddxcos⁡2x=ddxcos2⁡x-ddxsin2⁡x

By using power rule combined with chain rule and using rule(iv),

-2sin⁡2x=2.cos⁡x-sin⁡x-2.sin⁡xcos⁡x=-4.sin⁡xcos⁡x

sin⁡2x=2sin⁡xcos⁡x

Conclusion:

Double angle formula for sine function: sin⁡2x=2sin⁡xcos⁡x

To determine

(b)

To obtain:

Addition formula for cosine function.

Answer

cos⁡(x+a)=cos⁡a.cos⁡x-sin⁡a.sin⁡x

Explanation

1) Given:

sin⁡(x+a)=sin⁡xcos⁡a+cos⁡xsin⁡a

2) Calculation:

Differentiate given equation with respect to x,

ddxsin⁡(x+a)=ddx[sin⁡xcos⁡a+cos⁡xsin⁡a]

By using sum rule and using rule (iv),(v)

cos⁡x+a.ddxx+a=ddxsin⁡xcos⁡a+ddxcos⁡xsin⁡a

cos⁡(x+a)*1=cos⁡a.ddxsin⁡x+sin⁡a.ddxcos⁡x

cos⁡(x+a)=cos⁡a.cos⁡x+sin⁡a.-sin⁡x

cos⁡(x+a)=cos⁡a.cos⁡x-sin⁡a.sin⁡x

cos⁡(x+a)=cos⁡a.cos⁡x-sin⁡a.sin⁡x

Conclusion:

Addition formula for cosine function: cos⁡(x+a)=cos⁡a.cos⁡x-sin⁡a.sin⁡x