(sec(x) / (sec(x)-1)) - (sec(x) / (sex(x)+1)) = 2cot(x)csc(x)
On the left, get a common denominator of
(secx-1)(secx+1) giving
secx(secx+1)-secx(secx-1) all over the denominator
which reduces to
2secx over the denominator
2secx/(sec^2x-1)=
2secx*cos^2x/(1-cos^2x)=2cosx/sin^2x
=2ctnx/sinx=2ctnx*cscx
To prove this trigonometric equation, let's simplify both sides step by step:
Starting with the left-hand side (LHS):
LHS = (sec(x) / (sec(x) - 1)) - (sec(x) / (sec(x) + 1))
To simplify this expression, we'll find a common denominator for the fractions:
LHS = (sec(x) * (sec(x) + 1) - sec(x) * (sec(x) - 1)) / ((sec(x) - 1) * (sec(x) + 1))
Expanding the numerator:
LHS = (sec^2(x) + sec(x) - sec^2(x) + sec(x)) / ((sec^2(x) - 1))
The sec^2(x) terms cancel out:
LHS = (2sec(x)) / ((sec^2(x) - 1))
Now, we'll use the Pythagorean identity sec^2(x) - 1 = tan^2(x):
LHS = (2sec(x)) / (tan^2(x))
Recall the definition of cot(x) = 1/tan(x) and csc(x) = 1/sin(x):
LHS = (2sec(x)) / (cot^2(x))
Finally, remember the identity cot(x) = cos(x)/sin(x) and substitute it:
LHS = (2sec(x)) / ((cos(x)/sin(x))^2)
Simplifying this equation:
LHS = (2sec(x) * sin^2(x)) / cos^2(x)
We can now simplify the right-hand side (RHS) of the equation:
RHS = 2cot(x)csc(x)
= 2(cot(x) * csc(x))
= 2((cos(x)/sin(x)) * (1/sin(x)))
= 2(cos(x)/(sin^2(x)))
Now, let's manipulate the RHS to match the LHS:
RHS = 2(cos(x)/(sin^2(x)))
(2cos(x) * sin^2(x)) / cos^2(x)
By comparing both sides of the equation, we can conclude that:
LHS = RHS
Therefore, the equation is proved to be true.