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

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## Step-by-step Solution

Problem to solve:

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

Specify the solving method

1

Applying the trigonometric identity: $\cot\left(\theta\right)=\frac{1}{\tan\left(\theta\right)}$

$\tan\left(x\right)+\frac{1}{\tan\left(x\right)}=\sec\left(x\right)\csc\left(x\right)$
2

Combine all terms into a single fraction with $\tan\left(x\right)$ as common denominator

$\frac{\tan\left(x\right)\tan\left(x\right)+1}{\tan\left(x\right)}=\sec\left(x\right)\csc\left(x\right)$
3

When multiplying two powers that have the same base ($\tan\left(x\right)$), you can add the exponents

$\frac{\tan\left(x\right)^2+1}{\tan\left(x\right)}=\sec\left(x\right)\csc\left(x\right)$
4

Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$

$\frac{\sec\left(x\right)^2}{\tan\left(x\right)}=\sec\left(x\right)\csc\left(x\right)$
5

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

$\frac{\frac{1}{\cos\left(x\right)^2}}{\tan\left(x\right)}=\sec\left(x\right)\csc\left(x\right)$
6

Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$

$\frac{\frac{1}{\cos\left(x\right)^2}}{\frac{\sin\left(x\right)}{\cos\left(x\right)}}=\sec\left(x\right)\csc\left(x\right)$
7

Simplify the fraction

$\frac{\cos\left(x\right)}{\cos\left(x\right)^2\sin\left(x\right)}=\sec\left(x\right)\csc\left(x\right)$
8

Simplify the fraction by $\cos\left(x\right)$

$\frac{1}{\cos\left(x\right)\sin\left(x\right)}=\sec\left(x\right)\csc\left(x\right)$
9

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

$\frac{\sec\left(x\right)}{\sin\left(x\right)}=\sec\left(x\right)\csc\left(x\right)$
10

The reciprocal sine function is cosecant: $\frac{1}{\sin(x)}=\csc(x)$

$\sec\left(x\right)\csc\left(x\right)=\sec\left(x\right)\csc\left(x\right)$
11

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

true

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

### Main topic:

Trigonometric Identities

~ 0.07 s