Step-by-step Solution

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

Go!
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Final Answer

true

Step-by-step explanation

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

Multiplying the fraction by $-1$

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

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

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

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

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

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

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

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

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

Applying an identity of double-angle cosine: $\cos\left(2\theta\right)=1-2\sin\left(\theta\right)^2$

$\sec\left(x\right)=\frac{-\left(1-2\sin\left(x\right)^2\right)+2\cos\left(x\right)^2}{\cos\left(x\right)}$
7

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

$\sec\left(x\right)=\frac{-1+2\sin\left(x\right)^2+2\cos\left(x\right)^2}{\cos\left(x\right)}$
8

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

$\sec\left(x\right)=\frac{1}{\cos\left(x\right)}$
9

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

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

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

true

Final Answer

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

Related formulas:

1. See formulas

Time to solve it:

~ 0.08 s (SnapXam)