Final Answer
Step-by-step Solution
Problem to solve:
Solving method
Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$
Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$
Divide fractions $\frac{\frac{1}{\sin\left(x\right)}}{\cos\left(x\right)}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$
Multiply $\frac{1}{\sin\left(x\right)\cos\left(x\right)}$ by $\frac{sin(x)^2+cos(x)^2}{sin(x)^2+cos(x)^2}$
Multiplying fractions $\frac{1}{\sin\left(x\right)\cos\left(x\right)} \times \frac{\sin\left(x\right)^2+\cos\left(x\right)^2}{\sin\left(x\right)^2+\cos\left(x\right)^2}$
Applying the pythagorean identity: $\sin^2\left(\theta\right)+\cos^2\left(\theta\right)=1$
Split the fraction $\frac{\sin\left(x\right)^2+\cos\left(x\right)^2}{\sin\left(x\right)\cos\left(x\right)}$ in two fractions with common denominator $\sin\left(x\right)\cos\left(x\right)$
Simplify the fraction by $\sin\left(x\right)$
Simplify the fraction by $\cos\left(x\right)$
Apply the trigonometric identity: $\frac{\sin\left(x\right)}{\cos\left(x\right)}$$=\tan\left(x\right)$
Apply the trigonometric identity: $\frac{\cos\left(x\right)}{\sin\left(x\right)}$$=\cot\left(x\right)$
Rewrite $\frac{1}{\sin\left(x\right)\cos\left(x\right)}$ as $\tan\left(x\right)+\cot\left(x\right)$ by applying trigonometric identities
Since both sides of the equality are equal, we have proven the identity