anonymous 5 years ago Which of the following is equivalent to the expression below? 1+sec(theta) ----------------- tan(theta)+sin(theta)

1. anonymous

$\frac{1 + sec(\theta)}{tan(\theta) + sin(\theta)} = \frac{1 + \frac{1}{cos(\theta)}}{\frac{sin(\theta)}{cos(\theta)} + sin(\theta)}$

2. anonymous

I dunno what options you have, these trig identities can get complicated.

3. anonymous

csc(theta) tan(theta) -1 sec(theta) cot(theta)

4. anonymous

Well, from the denominator you can factor out a sin(theta) from each term.. what do you have then?

5. anonymous

uh (1+1/cos(theta))/(cos(theta))?

6. anonymous

no. try factoring again $\frac{sin\theta}{cos\theta} + sin\theta = ?$

7. anonymous

sec(theta)

8. anonymous

that's the answer yes, but not what I was asking.. I assume you saw the solution after you factored it?

9. anonymous

err actually no, that's not the answer. nvm

10. anonymous

what do you get when you factor out a sin(theta) from that expression in the denominator?

11. anonymous

uhhhh ok how do I factor it out?

12. anonymous

How do you factor out the a from 2a + ba ?

13. anonymous

Or factor out a 2 from (4 + 6a)

14. anonymous

sin(theta)(1/cos(theta)

15. anonymous

close

16. anonymous

try this one: factor completely (10x+2)

17. anonymous

2(5x+1)

18. anonymous

right. Now try $$\frac{a}{b} + a$$

19. anonymous

or we can write it as$\frac{1}{b}*a + a$

20. anonymous

if that makes it clearer

21. anonymous

sin(θ)(1+cos(θ))/cos(θ)

22. anonymous

no.

23. anonymous

$\frac{a}{b} + a = a(\frac{1}{b} + 1). a = sin(\theta), b = cos(\theta)$$\implies \frac{sin(\theta)}{cos(\theta)} + sin(\theta) = sin\theta(\frac{1}{cos\theta} + 1)$

24. anonymous

oh ok.... i guess i was kinda close, but not really

25. anonymous

so from there you should see something nice to cancel and get a simple answer

26. anonymous

what like the 1?

27. anonymous

no

28. anonymous

Like the [1 + 1/cos(theta)] you have on top and bottom

29. anonymous

what equation are you looking at?

30. anonymous

$\frac{1+\frac{1}{cos\theta}}{sin\theta (1 + \frac{1}{cos\theta})}$

31. anonymous

so all I have left is 1/sin(theta)?

32. anonymous

indeed

33. anonymous

which equals...csc(theta)

34. anonymous

indeed

35. anonymous

wow sorry about making that so difficult thank you so much!

36. anonymous

lol, np