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

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Solution

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Final answer to the problem

true

Step-by-step Solution

How should I solve this problem?

  • Prove from RHS (right-hand side)
  • Prove from LHS (left-hand side)
  • Express everything into Sine and Cosine
  • Exact Differential Equation
  • Linear Differential Equation
  • Separable Differential Equation
  • Homogeneous Differential Equation
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Load more...
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1

Starting from the right-hand side (RHS) of the identity

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Multiply $-1$ times $-1$

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

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

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

Cancel like terms $2\cos\left(x\right)^2$ and $-2\cos\left(x\right)^2$

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

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

$\sec\left(x\right)$
9

Since we have reached the expression of our goal, we have proven the identity

true

Final answer to the problem

true

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Plotting: $true$

Similar Problems Solved

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$sec\left(x\right)-tan\left(x\right)sin\left(x\right)=\frac{1}{sec\left(x\right)}$

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

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Prove the identity

$\sec x-\sin x\tan x=\cos x$

Prove the identity

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

Main Topic: Trigonometric Identities

In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables where both sides of the equality are defined.

Used Formulas

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