Step-by-step Solution

Prove the trigonometric identity $\cot\left(x\right)\sec\left(x\right)=\csc\left(x\right)$

Go!
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Final Answer

true

Step-by-step explanation

Problem to solve:

$\cot\left(x\right)\cdot\sec\left(x\right)=\csc\left(x\right)$

Choose the solving method

1

Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$

$\cot\left(x\right)\frac{1}{\cos\left(x\right)}=\csc\left(x\right)$
2

Multiply the fraction and term

$\frac{\cot\left(x\right)}{\cos\left(x\right)}=\csc\left(x\right)$
3

Apply the trigonometric identity: $\cot\left(x\right)$$=\frac{\cos\left(x\right)}{\sin\left(x\right)}$

$\frac{\frac{\cos\left(x\right)}{\sin\left(x\right)}}{\cos\left(x\right)}=\csc\left(x\right)$
4

Divide fractions $\frac{\frac{\cos\left(x\right)}{\sin\left(x\right)}}{\cos\left(x\right)}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$

$\frac{\cos\left(x\right)}{\sin\left(x\right)\cos\left(x\right)}=\csc\left(x\right)$
5

Simplify the fraction $\frac{\cos\left(x\right)}{\sin\left(x\right)\cos\left(x\right)}$ by $\cos\left(x\right)$

$\frac{1}{\sin\left(x\right)}=\csc\left(x\right)$
6

The reciprocal sine function is cosecant

$\csc\left(x\right)=\csc\left(x\right)$
7

Since both sides of the equality are equal, we have proven the identity

true

Final Answer

true
$\cot\left(x\right)\cdot\sec\left(x\right)=\csc\left(x\right)$

Related formulas:

1. See formulas

Time to solve it:

~ 0.04 s (SnapXam)