Prove the trigonometric identity $\frac{\sin\left(x+y\right)}{\cos\left(x\right)\cos\left(y\right)}=\tan\left(x\right)+\tan\left(y\right)$

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$\frac{\sin\left(x+y\right)}{\cos\left(x\right)\cos\left(y\right)}$

Learn how to solve problems step by step online. Prove the trigonometric identity sin(x+y)/(cos(x)cos(y))=tan(x)+tan(y). Starting from the left-hand side (LHS) of the identity. Using the sine of a sum formula: \sin(\alpha\pm\beta)=\sin(\alpha)\cos(\beta)\pm\cos(\alpha)\sin(\beta), where angle \alpha equals [x], and angle \beta equals [y]. Expand the fraction \frac{\sin\left(x\right)\cos\left(y\right)+\cos\left(x\right)\sin\left(y\right)}{\cos\left(x\right)\cos\left(y\right)} into 2 simpler fractions with common denominator \cos\left(x\right)\cos\left(y\right). Simplify the resulting fractions.