Final Answer
true
Step-by-step explanation
Problem to solve:
$\tan\left(x\right)+\cot\left(x\right)=\sec\left(x\right)\cdot\csc\left(x\right)$
Choose the solving method
1
Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$
$\tan\left(x\right)+\cot\left(x\right)=\frac{1}{\cos\left(x\right)}\csc\left(x\right)$
2
Multiply the fraction and term
$\tan\left(x\right)+\cot\left(x\right)=\frac{\csc\left(x\right)}{\cos\left(x\right)}$
3
Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$
$\tan\left(x\right)+\cot\left(x\right)=\frac{\frac{1}{\sin\left(x\right)}}{\cos\left(x\right)}$
4
Divide fractions $\frac{\frac{1}{\sin\left(x\right)}}{\cos\left(x\right)}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$
$\tan\left(x\right)+\cot\left(x\right)=\frac{1}{\sin\left(x\right)\cos\left(x\right)}$
5
Rewrite $\frac{1}{\sin\left(x\right)\cos\left(x\right)}$ as $\tan\left(x\right)+\cot\left(x\right)$ by applying trigonometric identities
$\tan\left(x\right)+\cot\left(x\right)=\tan\left(x\right)+\cot\left(x\right)$
6
Since both sides of the equality are equal, we have proven the identity
true
Final Answer
true