## sarah110128 2 years ago prove: (1+cosθ+cos2θ)/(sinθ+sin2θ)=cotθ

1. ajprincess

$$\cos2\theta=2\cos^2\theta-1$$ $$\sin2\theta=2\sin\theta\cos\theta$$ using these see if u can do. If u find any difficulties and need some more help plz ask:)

2. AnimalAin

$(1+\cosθ+\cos2θ)/(\sinθ+\sin2θ)=\cotθ \implies$$\cosθ+2\cos^2θ =\frac{\cos \theta}{\sin \theta}(\sin \theta+2 \sin \theta \cos \theta)=\cos \theta + 2 \cos^2 \theta$

3. sarah110128

prove: (1+cosθ+cos2θ)/(sinθ+sin2θ)=cotθ (1+cosθ+cos2θ)/(sinθ+2sinθcosθ) (cosθ+cos θ⁡ )/(sinθ+sinθ) cos2θ/sin2θ (2〖cos〗^2 θ-1)/2cosθsinθ (cosθ-1)/sinθ not sure where i went wrong... :/

4. ajprincess

Taking the expression in the left hand side $\frac{1+\cos\theta+\cos2\theta}{\sin\theta+\sin2\theta}$ $=\frac{1+\cos\theta+2\cos^2\theta-1}{\sin\theta+2\sin\theta\cos\theta}$ factor out $$\cos\theta$$ from numerator and $$\sin\theta$$ from denominator $=\frac{\cos\theta+2\cos^2\theta}{\sin\theta+2\sin\theta\cos\theta}$ $=\frac{\cos\theta(1+2\cos\theta)}{\sin\theta(1+2\cos\theta)}$ cancelling the common factor $$(1+2\cos\theta)$$ $=\frac{\cos\theta}{\sin\theta}$ $=\cot\theta$ nw we have shown that lhs=rhs hence proved

5. ajprincess

Is that clear @sarah11028

6. sarah110128

Sometimes I wonder why I'm even doing maths....hahaha thank you for you help! :)

7. ajprincess

welcome:) it is really easy if practice a bit:)

8. sarah110128

haahha lets hope it get easy! :) And hopefully it'll be my last year of maths

9. sarah110128

Cos3θ=4cos^3θ-3cosθ (I am trying to prove this identity) cos3θ = cos(θ+2θ) = cosθ cos2θ - sinθ sin2θ = cosθ (2cos^2θ - 1) - 2sin^2θ cosθ (basically here can you please explain how to go from sinθ sin2θ ->2sin^2θ cosθ ) = 2cos^3θ - cosθ - 2(1-cos^2θ) cosθ = 4cos^3θ - 3cosθ thanks

10. sarah110128

@ajprincess

11. ajprincess

$\sin2\theta=2\sin\theta\cos\theta$ $\sin2\theta*\sin\theta=2\sin\theta\cos\theta*\sin\theta$ $=2\sin\theta*\sin\theta*cos\theta$ $=2\sin^2\theta\cos\theta$ Is that clear?

12. ajprincess

$\sin2\theta=\sin(\theta+\theta)$ $=\sin\theta\cos\theta+\cos\theta\sin\theta$ $=2\sin\theta\cos\theta$

13. ajprincess

getting it @sarah110128?

14. sarah110128

ohh thanks, you've been a great help! :)

15. ajprincess

welcome:)