Step-by-Step Guide: Simplifying ln|csc(u) – cot(u)| + c in Math

ln|cscu-cotu|+c

The given expression is ln|csc(u) – cot(u)| + c

The given expression is ln|csc(u) – cot(u)| + c.

1. First, let’s simplify the expression inside the absolute value signs: csc(u) – cot(u).
– To simplify this expression, we need to recall the trigonometric identities:
– csc(u) = 1/sin(u)
– cot(u) = cos(u)/sin(u)
– Now we can substitute these identities into the given expression: 1/sin(u) – cos(u)/sin(u).
– Combining these fractions, we get (1 – cos(u))/sin(u).

2. Next, we take the natural logarithm (ln) of the absolute value of the expression we obtained in step 1: ln|(1 – cos(u))/sin(u)|.
– The natural logarithm ln is the inverse of the exponential function, and it gives us the power to which the base (e) must be raised to obtain a certain number.
– The absolute value |x| ensures that we consider both positive and negative values of x.

3. Finally, we add the constant c to the expression we obtained in step 2.

So, the simplified form of the given expression is ln|(1 – cos(u))/sin(u)| + c.

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