Final answer to the problem
$f\left(x\right)=\ln\left(1+\sqrt{\sin\left(x\right)}\right)-\ln\left(1-\sqrt{\sin\left(x\right)}\right)+2\arctan\left(\sqrt{\sin\left(x\right)}\right)$
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Step-by-step Solution
1
The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator
$f\left(x\right)=\ln\left(1+\sqrt{\sin\left(x\right)}\right)-\ln\left(1-\sqrt{\sin\left(x\right)}\right)+2\arctan\left(\sqrt{\sin\left(x\right)}\right)$
Final answer to the problem
$f\left(x\right)=\ln\left(1+\sqrt{\sin\left(x\right)}\right)-\ln\left(1-\sqrt{\sin\left(x\right)}\right)+2\arctan\left(\sqrt{\sin\left(x\right)}\right)$