Final Answer
true
Step-by-step Solution
Problem to solve:
$\cot\left(x\right)\sec\left(x\right)=\csc\left(x\right)$
Specify the solving method
1
Starting from the left-hand side (LHS) of the identity
$\cot\left(x\right)\sec\left(x\right)$
2
Apply the trigonometric identity: $\displaystyle\cot(x)=\frac{\cos(x)}{\sin(x)}$
$\frac{\cos\left(x\right)}{\sin\left(x\right)}\sec\left(x\right)$
Why does cot(x) = (cos(x)/(sin(x) ?
3
Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$
$\frac{\cos\left(x\right)}{\sin\left(x\right)}\frac{1}{\cos\left(x\right)}$
Multiplying fractions $\frac{\cos\left(x\right)}{\sin\left(x\right)} \times \frac{1}{\cos\left(x\right)}$
$\frac{\cos\left(x\right)}{\sin\left(x\right)\cos\left(x\right)}$
Simplify the fraction $\frac{\cos\left(x\right)}{\sin\left(x\right)\cos\left(x\right)}$ by $\cos\left(x\right)$
$\frac{1}{\sin\left(x\right)}$
4
Multiplying fractions $\frac{\cos\left(x\right)}{\sin\left(x\right)} \times \frac{1}{\cos\left(x\right)}$
$\frac{1}{\sin\left(x\right)}$
5
The reciprocal sine function is cosecant: $\frac{1}{\sin(x)}=\csc(x)$
$\csc\left(x\right)$
6
Since we have reached the expression of our goal, we have proven the identity
true
Final Answer
true