Final Answer
Step-by-step Solution
Problem to solve:
Solving method
Applying the trigonometric identity: $\cot\left(\theta\right)=\frac{1}{\tan\left(\theta\right)}$
Combine all terms into a single fraction with $\tan\left(x\right)$ as common denominator
When multiplying two powers that have the same base ($\tan\left(x\right)$), you can add the exponents
Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$
Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$
Divide fractions $\frac{\frac{1}{\cos\left(x\right)^2}}{\tan\left(x\right)}$ 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 tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$
Multiplying the fraction by $\cos\left(x\right)^2$
Simplify the fraction by $\cos\left(x\right)$
Apply trigonometric identities to simplify $\cos\left(x\right)^2\tan\left(x\right)$ into $\cos\left(x\right)\sin\left(x\right)$
Applying the trigonometric identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$
The reciprocal sine function is cosecant: $\frac{1}{\sin(x)}=\csc(x)$
Since both sides of the equality are equal, we have proven the identity
