Question

`( 1+ cot(\theta )- csc(\theta ) )( 1- tan(\theta )+ sec(\theta ) )= 2` ?

Answer 1

The solution to `Equation=2`
is
`theta=pi/2` (modulo `2 pi`).

HOWEVER, I REALLY THINK THERE IS A MISTAKE IN THE EQUATION AS THINGS WOULD CANCEL OUT NICELY IF THE EQUATION IS THE FOLLOWING:
`Equation=(1 + cot(theta) - csc(theta))(1+tan(theta)+sec(theta))`

(should be `+tan(theta)` instead of `-tan(theta)`).

Explanation:

Since the question is vague, I would just try to simplify the equation a bit first.
We start with:
`Equation=(1 + cot(theta) - csc(theta))(1-tan(theta)+sec(theta))`

and rewrite everything in terms of sine and cosine:
`=(1+cos(theta)/sin(theta) - 1/sin(theta)) (1-sin(theta)/cos(theta) + 1/cos(theta))`

Simplify a bit more:
`=(1+ (cos(theta) - 1)/(sin(theta))) (1+(1-sin(theta))/(cos(theta)))`

Opening up the parentheses:
`=1+(1-sin(theta))/(cos(theta))+(cos(theta) -1)/(sin(theta))+((cos(theta)-1)(1-sin(theta)))/(sin(theta)cos(theta))`

And putting everything under the same denominator gives:
`=(sin(theta)cos(theta)+sin(theta)(1-sin(theta))+cos(theta)(cos(theta)-1)+(cos(theta)-1)(1-sin(theta)))/(sin(theta) cos(theta))`

Now, we look just at the numerator:
`=sin(theta)cos(theta)+sin(theta)-sin^2(theta)+cos^2(theta)-cos(theta)+cos(theta)-sin(theta)cos(theta)-1+sin(theta)`
`=2sin(theta) -sin^2(theta) + cos^2(theta)-1`
`=2sin(theta)-sin^2(theta)+cos^2(theta)-sin^2(theta)-cos^2(theta)`
`=2sin(theta)(1-sin(theta))`

Now, back to the equation:
`Equation=(2sin(theta)(1-sin(theta)))/(sin(theta) cos(theta))`
So,
`Equation=2((1-sin(theta)))/(cos(theta))`

Now, let's solve for theta when `Equation =2`
This happens when `(1-sin(theta))/(cos(theta))=1`
i.e.
`1-sin(theta)=cos(theta)`
i.e.
`sin(theta)+cos(theta)=1`
This only happens when either `sin(theta)=1` and `cos(theta)=0`
when `theta=pi/2` (modulo `2pi`)
or when `sin(theta)=0` and `cos(theta)=1` when `theta=0` (modulo `2pi`).
Since `Equation =2((1-sin(theta)))/(cos(theta))`, `cos(theta)` cannot be 0 and
thus the solution to `Equation=2`
is
`theta=pi/2` (modulo `2 pi`).

HOWEVER, I REALLY THINK THERE IS A MISTAKE IN THE EQUATION AS THINGS WOULD CANCEL OUT NICELY IF THE EQUATION IS THE FOLLOWING:
`Equation=(1 + cot(theta) - csc(theta))(1+tan(theta)+sec(theta))`
(`+tan(theta)` instead of `-tan(theta)`).
Then, the equation is indeed 2 for all `theta`.