## anonymous one year ago Prove tan(θ / 2) = sin θ / (1 + cos θ) for θ in quadrant 1. I've looked at http://openstudy.com/study#/updates/4f78f650e4b0ddcbb89e7fe4 and http://openstudy.com/study#/updates/511ffe90e4b03d9dd0c5162b but I still can't figure it out!

1. anonymous

|dw:1440229429187:dw| I can get to here but I don't know how to prove it from this point on.

2. anonymous

Multiply numerator and denominator with sqrt( 1+cos theta). |dw:1440231020172:dw| simply this to get answer Source: http://www.mathskey.com/question2answer/

3. IrishBoy123

i haven't looked at the other thread but this seems the most obvious way to take it $$\large tan\frac{θ }{ 2} = \frac{sin θ}{ 1 + cos θ}$$ so you are proving $$\large \frac{sin\frac{θ }{ 2}}{cos\frac{θ }{ 2}} = \frac{2sin \frac{θ }{ 2} cos \frac{θ }{ 2}}{ 1 + (1 - 2 sin^2 \frac{θ }{ 2})}$$ now, simplify the RHS

4. IrishBoy123

as for the significance of Q1 |dw:1440235915730:dw|

5. anonymous

How do I simplify the right hand side? Sorry, I'm having a lot of trouble with this.

6. IrishBoy123

|dw:1440237322888:dw|

7. anonymous

$(2-2\sin^2 \theta)^2$ Is this right?

8. IrishBoy123

i will latex it so you see it clearly however, before doing so, i assume you know the double angle formula. that is how you break this down in the first place https://gyazo.com/943d7627c9c451b6b7653f69176505d6 yes? then, for the *denominator* $$1 + (1-2 sin^2 \frac{\theta}{2}) =$$ $$1 + 1-2 sin^2 \frac{\theta}{2} =$$ $$2-2 sin^2 \frac{\theta}{2} =$$ $$2(1- sin^2 \frac{\theta}{2}) = !!!$$

9. anonymous

cos(theta)+1 or $2 \cos^2(\theta/2)$ ?

10. IrishBoy123

https://gyazo.com/5d5b168435f599a323eefa614185c8ed $$\checkmark$$ now go back and compare numerator and denominator

11. IrishBoy123
12. anonymous

$(2\sin (\theta/2) \cos (\theta/2)) /2\cos^2(θ/2)$ $(\sin \theta/2)/(\cos \theta/2)$ $=\tan (\theta/2)$ Is this right?

13. IrishBoy123

$$\huge \checkmark$$

14. anonymous

Thank you so much! I finally feel like I get it now!

15. IrishBoy123

cool!