Start from the LHS (left-hand side)

Rewrite $\tan\left(x\right)$ in terms of sine and cosine

Rewrite $\cot\left(x\right)$ in terms of sine and cosine

Combine fractions with different denominator using the formula: $\displaystyle\frac{a}{b}+\frac{c}{d}=\frac{a\cdot d + b\cdot c}{b\cdot d}$

When multiplying two powers that have the same base ($\sin\left(x\right)$), you can add the exponents

When multiplying two powers that have the same base ($\cos\left(x\right)$), you can add the exponents

Applying the pythagorean identity: $\sin^2\left(\theta\right)+\cos^2\left(\theta\right)=1$

Start from the RHS (right-hand side)

Rewrite $\sec\left(x\right)$ in terms of sine and cosine

Rewrite $\csc\left(x\right)$ in terms of sine and cosine

Multiplying fractions $\frac{1}{\cos\left(x\right)} \times \frac{1}{\sin\left(x\right)}$

To prove an identity, we usually begin to work on the side of the equality that seems to be more complicated, or the side that is not expressed in terms of sine and cosine. In this problem, we will choose to work on the right side $\frac{1}{\cos\left(x\right)\sin\left(x\right)}$ to reach the left side $\frac{1}{\cos\left(x\right)\sin\left(x\right)}$

Since we have reached the expression of our goal, we have proven the identity