Use the properties of logarithms and trigonometric identities to verify the identity.
ln|cot x| = ln|cos x| - ln|sin x|
I have no idea where to begin with this one... your help is appreciated!
RS = ln (|cosx|/|sinx|)
= ln |cot x|
= LS
using cot Ø = cosØ/sinØ
and
log A - logB = log(A/B)
To verify the given identity, we will start by applying properties of logarithms and trigonometric identities. We will focus on simplifying the left-hand side (LHS) and the right-hand side (RHS) of the equation separately and show that they are equivalent.
Starting with the LHS:
ln|cot x|
We can rewrite cot x as cos x / sin x:
ln|cos x / sin x|
Now, applying the property of logarithms that states ln(a/b) = ln(a) - ln(b), we can split the logarithm:
ln|cos x| - ln|sin x|
This is the same as the RHS, so so far we have:
ln|cot x| = ln|cos x| - ln|sin x|
Now let's move to the RHS:
ln|cos x| - ln|sin x|
Using the property of logarithms, ln(a/b) = ln(a) - ln(b), we can combine the two logarithms:
ln|cos x / sin x|
We can rewrite cos x / sin x as cot x, so we have:
ln|cot x|
This brings us back to the LHS of the equation.
Since we have shown that the LHS is equal to the RHS, we have verified the given identity:
ln|cot x| = ln|cos x| - ln|sin x|
Therefore, the identity is verified using properties of logarithms and trigonometric identities.