What is 2ln|cot(x)|+2ln|sin(x)| ?

The solution to 2ln|cot(x)|+2ln|sin(x)| is ln(cos^2(x))

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simplify 2ln|cot(x)|+2ln|sin(x)|

simplify

2 ln ∣ cot (x) ∣ + 2 ln ∣ sin (x) ∣

ln (cos^{2} (x))

Solution steps

2 ln ∣ cot (x) ∣ + 2 ln ∣ sin (x) ∣

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

2 ln ∣ cot (x) ∣ = ln (∣ cot (x) ∣^{2})

= ln (∣ cot (x) ∣^{2}) + 2 ln ∣ sin (x) ∣

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

2 ln ∣ sin (x) ∣ = ln (∣ sin (x) ∣^{2})

= ln (∣ cot (x) ∣^{2}) + ln (∣ sin (x) ∣^{2})

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

ln (∣ cot (x) ∣^{2}) + ln (∣ sin (x) ∣^{2}) = ln (∣ cot (x) ∣^{2} ∣ sin (x) ∣^{2})

= ln (∣ cot (x) ∣^{2} ∣ sin (x) ∣^{2})

Simplify

∣ cot (x) ∣^{2} ∣ sin (x) ∣^{2} : cos^{2} (x)

= ln (cos^{2} (x))

3 lo g_{3} (2) + 2 lo g_{3} (4) - lo g_{3} (6)

2 lo g_{b} (x) + 9 lo g_{b} (z)

lo g_{5} (\frac{2 x - 1}{5})

4 ln (x) + 3 ln (y) - 6 ln (z)

lo g_{10} (\frac{x ^{3}}{z ^{2} y})

What is 2ln|cot(x)|+2ln|sin(x)| ?
The solution to 2ln|cot(x)|+2ln|sin(x)| is ln(cos^2(x))