# Tough Trig Question

1. Feb 23, 2010

### Hockeystar

1. The problem statement, all variables and given/known data
if tan(x) + cot(x) = 4 then what is

cos^2(x) + sin^2(x) + tan^2(x) + cot^2(x) + csc^2(x) + sec^2(x)

2. Relevant equations

cos^2(x) + sin^2(x) = 1
cot^2(x) = 1/(tan^2(x))

3. The attempt at a solution

= 1 + 14 + 1/(cos^2(x)sin^2(x))

Now I'm stuck.

2. Feb 23, 2010

### Staff: Mentor

How did you get this? "= 1 + 14 + 1/(cos^2(x)sin^2(x))"
What's the other side of this equation?

3. Feb 23, 2010

### Hockeystar

square first equation and you get tan^2(x) + cot^2(x) +2 = 16

4. Feb 23, 2010

### Staff: Mentor

You don't "square an equation;" you can square each side of an equation. Are you trying to say that (tan (x) + cot(x))2 = tan2(x) + cot2(x)?

How about (3 + 4)2? Is that equal to 32 + 42?

5. Feb 23, 2010

### TheoMcCloskey

Can we not solve for the first equation for $x$?

Express first equation in terms of $\sin(x)$ and $\cos(x)$, get common denominator, and solve for $x$.

Second equation can be simplified some-what by noting some additional trig identies for $\csc^2(x)$ and $\sec^2(x)$, but we can use solution for $x$ from first equation and substitute to solve for second equation directly.

6. Feb 24, 2010

### Staff: Mentor

There is no second equation. The idea is to simply evaluate that expression using the value(s) of x obtained from the first (and only) equation.

The first equation can be solved with minimal substitution, keeping things in terms of tan(x).

7. Feb 24, 2010

### vela

Staff Emeritus
No, he's saying

$$(\tan x+\cot x)^2=\tan^2 x+2 \tan x\cot x +\cot^2 x=\tan^2 x+2+\cot^2 x$$

since tan x cot x=1. Because the LHS is also equal to 42, he finds

$$\tan^2 x+\cot^2 x=16-2=14$$.

8. Feb 24, 2010

### vela

Staff Emeritus
Write this in terms of sin x and cos x and put the LHS over a common denominator. It'll simplify a bit, giving you a result that'll let you finish the problem.

9. Feb 24, 2010

### Mentallic

Take a look back at the original equality $tanx+cotx=4$ and note that $tanx=sinx/cosx$ and $cotx=cosx/sinx$

10. Feb 24, 2010

### Hockeystar

just solved it and I was so close. 1/(cos(x)sin(x)) = 4 Substitute that in for 1/(cos^2(x)sin^2(x)). You get 16. So the answer is 1 + 14 + 16 = 31

11. Feb 24, 2010

### Mentallic

Yeah, you were close :tongue:

12. Feb 24, 2010

### ƒ(x)

Lets not be hasty to bash other's ideas. The cot and tan cancel out, leaving 2.

13. Feb 24, 2010

### Staff: Mentor

I misread that final +2 on the left side in what Hockeystar posted, probably misreading it as the exponent on cot(x). My mistake.
"tan^2(x) + cot^2(x) +2 = 16"

14. Feb 24, 2010

### snshusat161

Given: tan(x) + cot (x) = 4

Square both side then,

$$tan^{2}x + cot^{2}x$$ + 2.tan(x) cot(x) = 16

but tan(x). cot(x) = 1

therefore, $$tan^2{x} + cot^2{x}$$ = 14.

Take this equation first.

15. Feb 24, 2010

### snshusat161

Now again consider tan(x) + cot(x) = 4

Now tan(x) = $$\frac{sin(x)}{cos(x)}$$ and similarly cot(x) = $$\frac{cos(x)}{sin(x)}$$

so

$$\frac{sin(x)}{cos(x)}$$ + $$\frac{cos(x)}{sin(x)}$$ = 4

$$\frac{sin^{2}x + cos^{2}x}{sin(x).cos(x)}$$ = 4

then it is equal to

$$\frac{1}{sin(x).cos(x)} = 4$$

take it equation second.

16. Feb 24, 2010

### snshusat161

$$cos^{2}(x) + sin^{2}(x) + tan^{2}(x) + cot^{2}(x) + csc^{2}(x) + sec^{2}(x)$$

Now here, $$cos^{2}(x) + sin^{2}(x)$$ = 1

$$tan^{2}(x) + cot^{2}(x)$$ = 14

17. Feb 24, 2010

### snshusat161

now last thing left is $$csc^{2}(x) + sec^{2}(x)$$

$$\frac{1}{sin^{2}x}$$ + $$\frac{1}{cos^{2}x}$$

that is equal to

$$\frac{cos^{2}x + sin^{2}x}{sin^2(x). cos^{2}x}$$

now look at equal second and I should leave rest up to you so that you can feel that you have done your home not me.

18. Feb 24, 2010

### snshusat161

now last thing left is $$csc^{2}(x) + sec^{2}(x)$$

$$\frac{1}{sin^{2}x}$$ + $$\frac{1}{cos^{2}x}$$

that is equal to

$$\frac{cos^{2}x + sin^{2}x}{sin^2(x). cos^{2}x}$$

now look at equal second and I should leave rest up to you so that you can feel that you have done your home not me.

19. Feb 25, 2010

### Mentallic

Oh wow, you gave such a subtle hint there... the OP would definitely feel like the homework was completed entirely on his/her own... :tongue:

By the way, the OP had all except the $tan^2x+cot^2x$ part figured out before even creating this thread and then did figure out the answer entirely shortly after. You would have known this if you read through the thread.

20. Feb 25, 2010

### snshusat161

Oh! I have seen only the last post and thought the question is still unsolved and so I've given the solution, but I think Mark got an answer that about the last step. Secondly I've not completed this question entirely cause I've seen somewhere in this forum that we have to not give complete solution for any problem. Is I'm right?