Trigonometry Help

Question

I have the answer to my question but I don't know how to get the answer myself so I figure try asking here.

 

I am given the equation 2/cot(x) - csc(x) and asked which of the following is equivalent to the expression, the answer is -2(cot(x) + cscx) but i'm not sure how this was solved.

Answer 1

 

      Those two expressions are not the same. Desmos.com shows 2 different graphs.

 

 
For example....
Let T = 30 degrees = pi/6

sin T = 1/2

cos T = sqrt(3)/2

tan T = sqrt(3)/3

cot T = 3/sqrt(3) = 3(sqrt(3)/3 = sqrt(3)

csc T = 2

2/cot(T) - csc(T) = 2 * tan T = csc(T) = 2*sqrt(3)/2 - 2 = sqrt(3) - 2

-2( cot(x) + csc(x)) = -2 ( sqrt(3) + 2) = -2 * sqrt(3) - 4

 

They are not the same.

 

Answer 2

It looks like parentheses are missing in the denominator on the left hand side, and maybe that is why you could not solve it.

 

2/[cot(x) - csc(x)] = ?

 

Multiply and divide by cot(x) + csc(x).

 

2/[cot(x) - csc(x)] =2/[cot(x) - csc(x)] {[cot(x) + csc(x)] / [cot(x) + csc(x)]}

 

= 2[cot(x) + csc(x)] / (cot²x - csc²x)

 

= 2[cot(x) + csc(x)] / (cos²x/sin²x - 1/sin²x) = 2[cot(x) + csc(x)] / [(cos²x - 1)/sin²x)]

 

= 2[cot(x) + csc(x)] / (-sin²x/sin²x) = -2[cot(x) + csc(x)]

Answer 3

I think your expression is:

 

2/(cot(x) - csc(x))

 

As you've written it, the csc(x) is not in the denominator, but I believe you intended it to be there. Multiply the expression by the conjugate of the denominator over itself:

 

2/(cot(x)-csc(x)) · (cot(x)+csc(x))/(cot(x)+csc(x)) = 2(cot(x)+csc(x))/(cot²(x)-csc²(x))

 

Now 1 + cot²(x) = csc²(x), so cot²(x) - csc²(x) = -1.  Substitute -1 is place of the (cot²(x)-csc²(x)) in the denominator:

 

2(cot(x)+csc(x))/(cot2(x)-csc2(x)) = 2(cot(x)+csc(x))/(-1) = -2(cot(x)+csc(x))