Here, we show you a step-by-step solved example of trigonometry. This solution was automatically generated by our smart calculator:
Starting from the left-hand side (LHS) of the identity
Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$
Combine all terms into a single fraction with $\cos\left(x\right)$ as common denominator
Divide fractions $\frac{1+\sec\left(x\right)}{\frac{\sin\left(x\right)\cos\left(x\right)+\sin\left(x\right)}{\cos\left(x\right)}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
Multiply the single term $\cos\left(x\right)$ by each term of the polynomial $\left(1+\sec\left(x\right)\right)$
Applying the trigonometric identity: $\cos\left(\theta \right)\sec\left(\theta \right) = 1$
Simplifying
Factoring by $\sin\left(x\right)$
Simplify the fraction $\frac{\cos\left(x\right)+1}{\sin\left(x\right)\left(1+\cos\left(x\right)\right)}$ by $\cos\left(x\right)+1$
Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$
Since we have reached the expression of our goal, we have proven the identity
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