# How to Solve Trig Identity Questions: Tips and Examples

• tg22542
In summary: Pythagorean identities......the other one being $$\sin^2 x + \cos^2 x = 1$$You write the left side of this as $$\sin^2 x = (1-\cos^2 x)$$You are effectively done!In summary, the student is practicing solving trigonometric identities and is seeking validation and assistance with two specific problems: sin2x = 2tanx/1+tan^2x and sin2x/sinx - cos2x/cosx = secx. They provide their attempts and ask for feedback and guidance on how to continue. Through the conversation
tg22542
Hey guys,I need some help on the following trig identities:

1) sin2x = 2tanx/1+tan^2x

2) sin2x/sinx - cos2x/cosx = secx

My attempts:

1) LS: sin2x
2sinxcosx

2sinx/cosx

2tanx/1+tanx

Not sure if this is right or not. I kind of understand my third step but it just doesn't seem right, any validations? If not, help please.

2) LS: sin2x/sinx - cos2x/cosx

2sinxcosx/sinx - ?

sinx/cosx - ?? i have no idea where to go from here

Do #1 in reverse, it should be easier, and take little steps, don't make your steps too large. Each step should be small enough to be obvious to the reader.

tg22542 said:
Hey guys,I need some help on the following trig identities:

1) sin2x = 2tanx/1+tan^2x

2) sin2x/sinx - cos2x/cosx = secx

My attempts:

1) LS: sin2x
2sinxcosx
Yes, sin(2x)= 2sin(x)cos(x)

2sinx/cosx
What? 2 sin(x)/cos(x)= 2 tan(x). Have you changed to the right side now??

2tanx/1+tanx
Where did this come from? It is what you want to arrive at but what are you doing to get it?

Not sure if this is right or not. I kind of understand my third step but it just doesn't seem right, any validations? If not, help please.

2) LS: sin2x/sinx - cos2x/cosx

2sinxcosx/sinx - ?

sinx/cosx - ?? i have no idea where to go from here

Frankly, I don't understand what you are trying to do. Please state exactly what you are doing to go from one step to another.

I just stated that I was trying to turn the left side into the right, sorry for the confusion. When I do these I just pick a side then use identities until it is equal to the opposite side.

hi tg22542!

(try using the X2 button just above the Reply box )
tg22542 said:
1) sin2x = 2tanx/1+tan^2x

2) sin2x/sinx - cos2x/cosx = secx

My attempts:

1) LS: sin2x
2sinxcosx

2sinx/cosx

2tanx/1+tanx

as verty says, it would be a lot easier to go from right to left!

if you must go from left to right, use cos = 1/sec
2) LS: sin2x/sinx - cos2x/cosx

first step is obviously to put them over a common denominator

okay so I believe I completed 1)

RS:

2tanx/1+tan^2x

2tanx/sec^2x

2sinx2cosx/sec^2x

2sinxcosx

sin2x

good? :)

thank you guys very much for your help so far, very appreciated

nevermind.. just realized the identity I used for sec is actually sec^2..

ugh

tg22542 said:
okay so I believe I completed 1)

RS:

2tanx/1+tan^2x

2tanx/sec^2x

2sinx2cosx/sec^2x

2sinxcosx

sin2x

good? :)

thank you guys very much for your help so far, very appreciated

Looks good to me.

For your other question, do you know any trig identities dealing with cos[2x]? Such as Cos[2x] = Cos[x]^2 - Sin[x]^2 ?

tg22542 said:
okay so I believe I completed 1)

RS:
The expressions below are all equal, so you should denote them as such by using =.

Also, what you wrote below needs parentheses around the terms in the denominator. What you wrote is really this:
$$\frac{2tan(x)}{1} + sec^2(x)$$
tg22542 said:
2tanx/1+tan^2x

2tanx/sec^2x

2sinx2cosx/sec^2x

2sinxcosx

sin2x

good? :)

thank you guys very much for your help so far, very appreciated

tg22542 said:
2tanx/1+tan^2x

2tanx/sec^2x

yes
2sinx2cosx/sec^2x

where did this come from?

use 1/sec = cos (as i said before)​

So could it go :

RS: 2tanx/1+tan^2x

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

(try using the X2 button just above the Reply box )
tg22542 said:
RS: 2tanx/1+tan^2x

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

i honestly have no idea how you got that

tg22542, in the exam, you must do one step at a time …

then the examiner can see what you're doing, and why, and at least give you some marks for getting something right!

try again, slowwwwly

RS: 2tanx/1+tan^2x

So I first used the identity tan^2x + 1 = sec^2x

So

= 2tanx + sec^2x

Next I used tanx=sinx/cosx

So:

2sinx/2cosx + sec^2x

Now I'm lost, It must equal 2sinx so I clearly have to get the cos to cancel out somehow, but I don't know which identities to use

tg22542 said:
RS: 2tanx/1+tan^2x

So I first used the identity tan^2x + 1 = sec^2x

correct

(except it would be clearer if you used brackets: 2tanx/(1+tan^2x) …

i suspect that's partly the reason why your next step was wrong …

as Mark44 has already said, always use brackets (parentheses))

So

= 2tanx + sec^2x

NO!

= 2tanx/sec2x …

carry on from there

Any idea on which identity to use from these point on? I'm lost :(

tg22542 said:
Any idea on which identity to use from these point on? I'm lost :(

What is tanx equal to? What is sec2x equal to? Now sub in.

=(2sinx/2cosx)/(1/cos^2x) ?

tg22542 said:
=(2sinx/2cosx)/(1/cos^2x) ?

yes

(except for one silly mistake )

and then?​

I have no idea, I feel likei cou,d multiply the top by 2cos leaving me 2sinx2cosx/cos which wou,d give me the answer? Correct?

tg22542 said:
I have no idea, I feel likei cou,d multiply the top by 2cos leaving me 2sinx2cosx/cos which wou,d give me the answer? Correct?

tg22542, i don't understand your defeatism

you say "i have no idea", even though you obviously do have an idea (and it's the correct one)

yes, your 1/(1/something) can obviously be canceled out

it's your "2"s that need a second look

(a/b)/(c/d) = (ad/bc) and then all you have to do is remember one more trig id. It would also be beneficial to not write 2tan(x) as (2sinx)/(2cosx), because that clearly isn't right.

I seriously don't know where to go from here..haha I'm just getting more and more confused

tg22542 said:
I seriously don't know where to go from here..haha I'm just getting more and more confused

You are really there, all you need to correct is your statement that $$2\tan x = \frac{2 \sin x}{2 \cos x}$$ Correct this and you are effectively done.

Oh okay so

(2sinx/2cosx)/(1/cos^2x)

2cosx and cos^2x cancel leaving me with 2sinx

:) done

tg22542 said:
Oh okay so

(2sinx/2cosx)/(1/cos^2x)

2cosx and cos^2x cancel leaving me with 2sinx

:) done

No. They don't cancel and, besides, what you want is sin2x, not 2sinx.

What is tanx equal to? Then what is 2 multiplied by tanx?

2sinx/2cosx?

tg22542 said:
2sinx/2cosx?

No. You figured that $$\tan x = \frac{\sin x}{\cos x}$$ What happens when you multiply this by 2?

You should know that $$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$$

Just to give you a concrete example. If you have 2*(4/2) that wouldn't be the same as (2*4)/(2*2), so why do you think 2(sinx/cosx) is the same as (2sinx)/(2cosx)?

tg22542 said:
2sinx/2cosx?

tg22542, how would you simplify 2sinx/2cosx?

## 1. How do I know which trig identities to use in a problem?

It is important to have a strong understanding of the basic trig identities, such as the Pythagorean identities and the double angle identities. These will often be used in more complex problems. Additionally, look for patterns and relationships between the given expressions to determine which identities may be useful.

## 2. What are some tips for simplifying trig identities?

One helpful tip is to work with one side of the identity at a time, using algebraic manipulation and substitution to simplify it. It may also be helpful to rewrite trigonometric functions in terms of sine and cosine, as these identities are more commonly used. Finally, be sure to keep track of any negative signs and use the properties of even and odd functions to simplify the expressions.

## 3. How can I check if my answer to a trig identity problem is correct?

The most reliable way to check if your answer is correct is to substitute it back into the original expression and see if both sides are equal. Additionally, you can use trig identities or a graphing calculator to verify your answer.

## 4. Are there any common mistakes to avoid when solving trig identities?

One common mistake is to only focus on one side of the identity and forget to simplify the other side. It is also important to be careful with signs and to avoid making arithmetic errors. Lastly, make sure to use the correct identities and not mix them up.

## 5. Can you provide an example of solving a trig identity problem?

One example is to prove the identity: sin^2(x) + cos^2(x) = 1. To solve this, we can use the Pythagorean identity: sin^2(x) + cos^2(x) = 1. Therefore, the identity is proven.

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