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 trigonometric 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

Combine fractions with different denominator using the formula: $\displaystyle\frac{a}{b}+\frac{c}{d}=\frac{a\cdot d + b\cdot c}{b\cdot d}$

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

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

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

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

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

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

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

Simplify $\frac{1}{\cos\left(x\right)\sin\left(x\right)}$ into $\frac{1}{\cos\left(x\right)}\csc\left(x\right)$ by applying trigonometric identities

$\frac{1}{\cos\left(x\right)}\csc\left(x\right)=\sec\left(x\right)\csc\left(x\right)$
8

Multiply the fraction and term

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

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

$\csc\left(x\right)\sec\left(x\right)=\sec\left(x\right)\csc\left(x\right)$
10

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:

1. See formulas

Time to solve it:

~ 0.05 s (SnapXam)