Verify the identity ln|cot(x) + tan(x)| = ln|sec(x)| + ln|csc (x)|
cotx + tanx = cosx/sinx + sinx/cosx = (cos^2x + sin^2x)/(sinx cosx)
= 1/(sinx cosx) = cscx * secx
To verify the given identity: ln|cot(x) + tan(x)| = ln|sec(x)| + ln|csc(x)|, we need to show that the expression on the left side of the equation is equal to the expression on the right side for all valid values of x.
Let's start by examining each side of the equation individually.
Left side: ln|cot(x) + tan(x)|
To simplify this expression, we can use the logarithmic properties and the trigonometric identities. First, we notice that the absolute value signs are redundant in this case since both cot(x) and tan(x) are always positive or always negative in the same interval.
Using the identity cot(x) = 1/tan(x), the left side becomes:
ln|cot(x) + tan(x)| = ln|1/tan(x) + tan(x)|
Next, we can combine the fractions under the logarithm using a common denominator:
ln|cot(x) + tan(x)| = ln|(1 + tan^2(x))/tan(x)|
Using the Pythagorean identity tan^2(x) + 1 = sec^2(x):
ln|cot(x) + tan(x)| = ln|(sec^2(x))/tan(x)|
Using the identity sec^2(x) = 1 + tan^2(x):
ln|cot(x) + tan(x)| = ln|(1 + tan^2(x))/tan(x)|
We can simplify further by multiplying the numerator and denominator by cos^2(x):
ln|cot(x) + tan(x)| = ln|((1 + tan^2(x))/tan(x))*(cos^2(x)/cos^2(x))|
Using the identity tan(x)/cos(x) = sin(x)/cos^2(x) = csc(x)/cos(x):
ln|cot(x) + tan(x)| = ln|(csc(x) + sin(x))/cos(x)|
Now, let's move on to the right side of the equation.
Right side: ln|sec(x)| + ln|csc(x)|
Using the identities sec(x) = 1/cos(x) and csc(x) = 1/sin(x), we have:
ln|sec(x)| + ln|csc(x)| = ln|(1/cos(x))| + ln|(1/sin(x))|
Using the logarithmic property ln(a) + ln(b) = ln(ab):
ln|sec(x)| + ln|csc(x)| = ln|(1/cos(x))*(1/sin(x))|
Combining the fractions, we get:
ln|sec(x)| + ln|csc(x)| = ln|1/(cos(x)*sin(x))|
Finally, using the reciprocal trigonometric identity cos(x)*sin(x) = sin(x)/cos(x) = tan(x):
ln|sec(x)| + ln|csc(x)| = ln|1/tan(x)|
Now, comparing both sides of the equation:
ln|cot(x) + tan(x)| = ln|sec(x)| + ln|csc(x)|
We can see that the expressions on the left and right sides of the equation are equivalent.
Therefore, we have verified the given identity: ln|cot(x) + tan(x)| = ln|sec(x)| + ln|csc(x)| for all valid values of x.