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

Step-by-step Solution

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Solution

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Final Answer

true

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 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

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

$\tan\left(x\right)+\cot\left(x\right)=\frac{1}{\cos\left(x\right)}\frac{1}{\sin\left(x\right)}$
3

Multiplying fractions $\frac{1}{\cos\left(x\right)} \times \frac{1}{\sin\left(x\right)}$

$\tan\left(x\right)+\cot\left(x\right)=\frac{1}{\cos\left(x\right)\sin\left(x\right)}$

Multiply $\frac{1}{\cos\left(x\right)\sin\left(x\right)}$ by $\frac{sin(x)^2+cos(x)^2}{sin(x)^2+cos(x)^2}$

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

Multiplying fractions $\frac{1}{\cos\left(x\right)\sin\left(x\right)} \times \frac{\sin\left(x\right)^2+\cos\left(x\right)^2}{\sin\left(x\right)^2+\cos\left(x\right)^2}$

$\frac{\sin\left(x\right)^2+\cos\left(x\right)^2}{\cos\left(x\right)\sin\left(x\right)\left(\sin\left(x\right)^2+\cos\left(x\right)^2\right)}$

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

$\frac{\sin\left(x\right)^2+\cos\left(x\right)^2}{\cos\left(x\right)\sin\left(x\right)}$

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

$\frac{1}{\cos\left(x\right)\sin\left(x\right)}$
4

Rewrite $\frac{1}{\cos\left(x\right)\sin\left(x\right)}$ as $\tan\left(x\right)+\cot\left(x\right)$ by applying trigonometric identities

$\tan\left(x\right)+\cot\left(x\right)=\tan\left(x\right)+\cot\left(x\right)$
5

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

true

Final Answer

true

Similar Problems

Prove the identity

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

Prove the identity

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

Prove the identity

$\tan\left(x\right)\cdot \cos\left(x\right)\cdot \csc\left(x\right)=1$

Prove the identity

$\sin\left(x\right)^2+\cos\left(x\right)^2=1$

Prove the identity

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

Prove the identity

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

Prove the identity

$\frac{1-\sin\left(x\right)}{\cos\left(x\right)}=\frac{\cos\left(x\right)}{1+\sin\left(x\right)}$
$\tan\left(x\right)+\cot\left(x\right)=\sec\left(x\right)\cdot\csc\left(x\right)$

Main topic:

Trigonometric Identities

Used formulas:

3. See formulas

Time to solve it:

~ 0.04 s

Related topics:

Trigonometric IdentitiesProving Trigonometric Identities

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