Proving Identities and Double angles problems

[sens_freak_18]

I've been working at proving these identites and I just can't seem to figure them out, some of them just come to me others I work on them for 10 mintues or more and just get more and more bogged down.

1) csc2x + cot2x = cotx so far i have
1/sin2x + cos2x/sin2x = cosx/sinx
1+cos2x/sinx = cosx/sinx

2) cos^2x + 4cosx + 3 = cosx + 3 so far i have
4cos^2x + 3 = cosx + 3

3)1+cosx/1-cosx - 1-cosx/1+cosx = 4cotxcscx so far i have
1+cosx/sin^2x - sin^2x/1+cosx = 4cosx/4sinx*1/sinx

4)cos^2x-csc^2x/cot^2x = sin^2x - sec^2x so far i have
cos^2x - 1/sin^2x/ cos^2X/sin^2x
cos^2x - 1/sin^2x * sin^2X/cos^2x
cos^2x - 1/cos^2x
cos^2x - sec^2x

5) 1 + sec^2x + cot^2x/ csc^2x = sec^2x

1+ 1/cos^2x + cos^2x/sin^2x/ 1/sin^2x = 1/cos^2x

6) sin(x+y) + sin(x-y)/cos(x+y) + cos(x-y) = tanx
I have absolutely no clue what to do with this one, I've tried changing the left side to cosxsiny+sinxcosy and stuff like that but it just doesn't come to me

Thanks for any help given to me, I really need some and greatly appreciate it!

 

[Hurkyl]

First off, you need to learn how to use parentheses correctly:

$$a + b / c + d = a + \frac{b}{c} + d$$

but

$$(a + b) / (c + d) = \frac{a + b}{c + d}$$


Secondly, I see you're converting everything into sines and cosines: that's usually a good thing.

But you're missing two other easy steps:

(1) Clearing denominators.

For example, in order to prove

$$\frac{a}{b} = \frac{c}{d}$$

whenever it is defined all you have to do is to prove

$$ad = bc$$

whenever both b and d are nonzero. (it's okay to prove this holds even when b or d are zero)

(2) Working with only one angle.

In a lot of these, I see that you have trig functions with different arguments. E.G. in the first one, cos x and cos 2x both appear. You can often fix that.

 

[Architeuthis Dux]

I can understand the frustration and difficulty in proving identities and solving double angle problems. These types of problems require a lot of practice and understanding of the fundamental trigonometric identities. It is important to approach these problems systematically and use known identities to simplify the given expressions.

For problem 1, you are on the right track. You can continue by using the identity: cos2x = 1 - sin2x. This will help you simplify the expression and reach the desired result.

For problem 2, try using the identity: cos2x = 1 - sin2x and then factor the resulting expression to reach the desired result.

For problem 3, use the identity: 1 + cosx = 2cos^2x/2 to simplify the expression and reach the desired result.

For problem 4, try using the identities: cos^2x = 1 - sin^2x and csc^2x = 1 + cot^2x to simplify the expression and reach the desired result.

For problem 5, use the identity: sec^2x = 1 + tan^2x to simplify the expression and reach the desired result.

For problem 6, you are correct in trying to use the sum and difference identities for sine and cosine. You can also use the identity: tanx = sinx/cosx to simplify the expression and reach the desired result.

Remember to always start by simplifying the given expression using known identities and then use algebraic manipulation to reach the desired result. Don't get discouraged if some problems take longer to solve, it just takes practice and patience. Keep up the good work!