___1____ __ ___1_____ = -2tan(x)sec(x)

1-cosecx 1+cosecx

Iwill read that as

1/(1 - cscx) - 1/(1 + cscx) = -2tanx secx

LS = (1 + cscx - (1 - cscx))/(1 - csc^2 x)
= 2cscx/(-cot^2 x)
= -2 (1/sinx)(tan^2 x)
= -2(1/sinx)(sin^2 x/cos^2 x)
= -2 cscx/cos^2 x
= -2 cscx sec^2 x
= RS

1/(1-csc) - 1/(1+csc)

= ((1+csc)-(1-csc))/(1-csc^2)
= 2csc/cot^2
= 2csc tan^2
= 2/sin sin^2/cos^2
= 2sin/cos^2
= 2tan sec

I seem to have lost a - sign. Hard to tell just what the original equation was.

Oops. I see where. Reiny did it right.

To simplify this expression, we can start by simplifying the numerator and denominator separately. Let's begin with the numerator.

If we multiply -2tan(x)sec(x) by 1-cosecx, we can distribute the -2 to both terms:

-2tan(x)sec(x)(1 - cosecx) = -2tan(x)sec(x) + 2tan(x)sec(x)cosecx

Now let's simplify the denominator.

To simplify 1 - cosecx, we can use the formula for the secant function: sec^2(x) = 1 + tan^2(x). We can rewrite this as:

1 - cosecx = 1 - (1/sinx) = (sinx/sinx) - (1/sinx) = (sinx - 1)/sinx

Now, we can substitute the simplified numerator and denominator back into the original expression:

(-2tan(x)sec(x) + 2tan(x)sec(x)cosecx) / ((sinx - 1)/sinx)

Next, we can simplify further by multiplying the numerator by sinx/sinx, which is equivalent to multiplying the entire expression by 1:

((-2tan(x)sec(x) + 2tan(x)sec(x)cosecx) / ((sinx - 1)/sinx)) * (sinx/sinx)

Simplifying this expression:

= (-2tan(x)sec(x)sinx + 2tan(x)sec(x)cosecxsinx) / (sinx - 1)

Finally, we can further simplify the numerator by factoring out the common term tan(x)sec(x)sinx:

= (-2tan(x)sec(x)sinx(1 - cosecx)) / (sinx - 1)

The final simplified expression is:

-2tan(x)sec(x)sinx(1 - cosecx) / (sinx - 1)