Final Answer
Step-by-step explanation
Problem to solve:
Choose the solving method
Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$
Apply the identity: $\cot\left(x\right)$$=\frac{\cos\left(x\right)}{\sin\left(x\right)}$
Apply the identity: $\frac{\sin\left(x\right)}{\cos\left(x\right)}$$=\tan\left(x\right)$
The reciprocal sine function is cosecant
Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$
Combine $\frac{\sin\left(x\right)}{\cos\left(x\right)}+\cos\left(x\right)\csc\left(x\right)$ in a single fraction
When multiplying two powers that have the same base ($\cos\left(x\right)$), you can add the exponents
Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$
Multiply the fraction and term
Combine $\sin\left(x\right)+\frac{\cos\left(x\right)^2}{\sin\left(x\right)}$ in a single fraction
When multiplying two powers that have the same base ($\sin\left(x\right)$), you can add the exponents
Applying the pythagorean identity: $\sin^2\left(\theta\right)+\cos^2\left(\theta\right)=1$
Divide fractions $\frac{\frac{1}{\sin\left(x\right)}}{\cos\left(x\right)}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$
Applying the trigonometric identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$
Multiply the fraction and term
The reciprocal sine function is cosecant
Since both sides of the equality are equal, we have proven the identity