Exercise

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

Step-by-step Solution

I. Express the LHS in terms of sine and cosine and simplify

1

Start from the LHS (left-hand side)

$\sec\left(x\right)$
2

Rewrite $\sec\left(x\right)$ in terms of sine and cosine

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

II. Express the RHS in terms of sine and cosine and simplify

3

Start from the RHS (right-hand side)

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

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

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

Simplify $\sin\left(x\right)\cos\left(x\right)$ using the trigonometric identity: $\sin(2x)=2\sin(x)\cos(x)$

$\frac{\sin\left(2x\right)\cos\left(x\right)-\cos\left(2x\right)\sin\left(x\right)}{\frac{\sin\left(2x\right)}{2}}$
6

Divide fractions $\frac{\sin\left(2x\right)\cos\left(x\right)-\cos\left(2x\right)\sin\left(x\right)}{\frac{\sin\left(2x\right)}{2}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$

$\frac{2\left(\sin\left(2x\right)\cos\left(x\right)-\cos\left(2x\right)\sin\left(x\right)\right)}{\sin\left(2x\right)}$
7

Multiply the single term $2$ by each term of the polynomial $\left(\sin\left(2x\right)\cos\left(x\right)-\cos\left(2x\right)\sin\left(x\right)\right)$

$\frac{2\sin\left(2x\right)\cos\left(x\right)-2\cos\left(2x\right)\sin\left(x\right)}{\sin\left(2x\right)}$
8

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

$\frac{4\sin\left(x\right)\cos\left(x\right)\cos\left(x\right)-2\cos\left(2x\right)\sin\left(x\right)}{\sin\left(2x\right)}$
Why does sin(2x) = 2sin(x)cos(x) ?
9

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

$\frac{4\sin\left(x\right)\cos\left(x\right)^2-2\cos\left(2x\right)\sin\left(x\right)}{\sin\left(2x\right)}$
10

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

$\frac{4\sin\left(x\right)\cos\left(x\right)^2-2\cos\left(2x\right)\sin\left(x\right)}{2\sin\left(x\right)\cos\left(x\right)}$
Why does sin(2x) = 2sin(x)cos(x) ?
11

Factor the polynomial $4\sin\left(x\right)\cos\left(x\right)^2-2\cos\left(2x\right)\sin\left(x\right)$ by it's greatest common factor (GCF): $2\sin\left(x\right)$

$\frac{2\sin\left(x\right)\left(2\cos\left(x\right)^2-\cos\left(2x\right)\right)}{2\sin\left(x\right)\cos\left(x\right)}$
12

Simplify the fraction

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

III. Choose what side of the identity are we going to work on

13

To prove an identity, we usually begin to work on the side of the equality that seems to be more complicated, or the side that is not expressed in terms of sine and cosine. In this problem, we will choose to work on the right side $\frac{2\cos\left(x\right)^2-\cos\left(2x\right)}{\cos\left(x\right)}$ to reach the left side $\frac{1}{\cos\left(x\right)}$

$\frac{1}{\cos\left(x\right)}=\frac{2\cos\left(x\right)^2-\cos\left(2x\right)}{\cos\left(x\right)}$
14

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

$\frac{2\cos\left(x\right)^2-\left(2\cos\left(x\right)^2-1\right)}{\cos\left(x\right)}$
15

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

$\frac{2\cos\left(x\right)^2-2\cos\left(x\right)^2- -1}{\cos\left(x\right)}$
16

Multiply $-1$ times $-1$

$\frac{2\cos\left(x\right)^2-2\cos\left(x\right)^2+1}{\cos\left(x\right)}$
17

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

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

IV. Check if we arrived at the expression we wanted to prove

18

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

true

Final answer to the exercise

true

Try other ways to solve this exercise

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