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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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Final Answer

true

Step-by-step Solution

Specify the solving method

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

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

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

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

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

true

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Prove from LHS (left-hand side)Express everything into Sine and Cosine

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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

2. See formulas

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