___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)