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Prove the trigonometric identity $\sec\left(x\right)=\frac{\sin\left(2x\right)}{\sin\left(x\right)}-\frac{\cos\left(2x\right)}{\cos\left(x\right)}$

Step-by-step Solution

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Solution

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Final Answer

true

Step-by-step Solution

Problem to solve:

$\sec\left(x\right)=\frac{\sin\left(2x\right)}{\sin\left(x\right)}-\frac{\cos\left(2x\right)}{\cos\left(x\right)}$

Choose the solving method

1

Using the sine double-angle identity: $\sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right)$

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

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

$\sec\left(x\right)=2\cos\left(x\right)+\frac{-\cos\left(2x\right)}{\cos\left(x\right)}$
3

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

$\sec\left(x\right)=\frac{2\cos\left(x\right)\cos\left(x\right)-\cos\left(2x\right)}{\cos\left(x\right)}$
4

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

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

Apply the trigonometric identity: $\cos\left(2x\right)$$=2\cos\left(x\right)^2-1$

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

Solve the product $-(2\cos\left(x\right)^2-1)$

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

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

$\sec\left(x\right)=\sec\left(x\right)$
8

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

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

Main topic:

Trigonometric Identities

Related Formulas:

2. See formulas

Time to solve it:

~ 0.11 s

Related topics:

Trigonometric IdentitiesProving Trigonometric IdentitiesFactorization

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