Step-by-step Solution

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

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Final Answer

true

Step-by-step Solution

Problem to solve:

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

Choose 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: $\sin^2\left(\theta\right)+\cos^2\left(\theta\right)=1$

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

Split the fraction $\frac{\sin\left(x\right)^2+\cos\left(x\right)^2}{\cos\left(x\right)\sin\left(x\right)}$ in two fractions with common denominator $\cos\left(x\right)\sin\left(x\right)$

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

Simplify the fraction $\frac{\sin\left(x\right)^2}{\cos\left(x\right)\sin\left(x\right)}$ by $\sin\left(x\right)$

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

Simplify the fraction $\frac{\cos\left(x\right)^2}{\cos\left(x\right)\sin\left(x\right)}$ by $\cos\left(x\right)$

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

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

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

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

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

Related Formulas:

3. See formulas

Time to solve it:

~ 0.08 s