Final Answer
Step-by-step Solution
Problem to solve:
Solving method
Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$
Apply the trigonometric identity: $\cot(x)=\frac{\cos(x)}{\sin(x)}$
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: $\sin^2\left(\theta\right)+\cos^2\left(\theta\right)=1$
The reciprocal sine function is cosecant: $\frac{1}{\sin(x)}=\csc(x)$
Applying the trigonometric identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$
Since both sides of the equality are equal, we have proven the identity