Step-by-step Solution

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

Go!
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Final Answer

true

Step-by-step explanation

Problem to solve:

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

Choose the solving method

1

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

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

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

$\frac{\sin\left(x\right)}{\cos\left(x\right)}+\frac{\cos\left(x\right)}{\sin\left(x\right)}=\sec\left(x\right)\csc\left(x\right)$
3

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

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

The reciprocal sine function is cosecant

$\tan\left(x\right)+\cos\left(x\right)\csc\left(x\right)=\sec\left(x\right)\csc\left(x\right)$
5

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

$\frac{\sin\left(x\right)}{\cos\left(x\right)}+\cos\left(x\right)\csc\left(x\right)=\sec\left(x\right)\csc\left(x\right)$
6

Combine $\frac{\sin\left(x\right)}{\cos\left(x\right)}+\cos\left(x\right)\csc\left(x\right)$ in a single fraction

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

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

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

Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$

$\frac{\sin\left(x\right)+\cos\left(x\right)^2\frac{1}{\sin\left(x\right)}}{\cos\left(x\right)}=\sec\left(x\right)\csc\left(x\right)$
9

Multiply the fraction and term

$\frac{\sin\left(x\right)+\frac{\cos\left(x\right)^2}{\sin\left(x\right)}}{\cos\left(x\right)}=\sec\left(x\right)\csc\left(x\right)$
10

Combine $\sin\left(x\right)+\frac{\cos\left(x\right)^2}{\sin\left(x\right)}$ in a single fraction

$\frac{\frac{\cos\left(x\right)^2+\sin\left(x\right)\sin\left(x\right)}{\sin\left(x\right)}}{\cos\left(x\right)}=\sec\left(x\right)\csc\left(x\right)$
11

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

$\frac{\frac{\cos\left(x\right)^2+\sin\left(x\right)^2}{\sin\left(x\right)}}{\cos\left(x\right)}=\sec\left(x\right)\csc\left(x\right)$
12

Applying the pythagorean identity: $\sin^2\left(\theta\right)+\cos^2\left(\theta\right)=1$

$\frac{\frac{1}{\sin\left(x\right)}}{\cos\left(x\right)}=\sec\left(x\right)\csc\left(x\right)$
13

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

$\frac{1}{\sin\left(x\right)\cos\left(x\right)}=\sec\left(x\right)\csc\left(x\right)$
14

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

$\frac{1}{\sin\left(x\right)}\sec\left(x\right)=\sec\left(x\right)\csc\left(x\right)$
15

Multiply the fraction and term

$\frac{\sec\left(x\right)}{\sin\left(x\right)}=\sec\left(x\right)\csc\left(x\right)$
16

The reciprocal sine function is cosecant

$\sec\left(x\right)\csc\left(x\right)=\sec\left(x\right)\csc\left(x\right)$
17

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

true

Final Answer

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

Related formulas:

2. See formulas

Steps:

17

Time to solve it:

~ 0.06 s (SnapXam)