Final answer to the problem
Step-by-step Solution
Specify the solving method
I. Express the LHS in terms of sine and cosine and simplify
Start from the LHS (left-hand side)
Rewrite $\sec\left(x\right)$ in terms of sine and cosine
II. Express the RHS in terms of sine and cosine and simplify
Start from the RHS (right-hand side)
Nothing to do here. The expression is already in terms of sine and cosine and simplified
III. Choose what side of the identity are we going to work on
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 $\frac{\sin\left(2x\right)}{\sin\left(x\right)}+\frac{-\cos\left(2x\right)}{\cos\left(x\right)}$ to reach the left side $\frac{1}{\cos\left(x\right)}$
Combine fractions with different denominator using the formula: $\displaystyle\frac{a}{b}+\frac{c}{d}=\frac{a\cdot d + b\cdot c}{b\cdot d}$
Simplify $\sin\left(x\right)\cos\left(x\right)$ using the trigonometric identity: $\sin(2x)=2\sin(x)\cos(x)$
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: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
Apply the trigonometric identity: $\sin\left(x\right)\cos\left(y\right)$$=\frac{\sin\left(x+y\right)+\sin\left(x-y\right)}{2}$
Simplify the product $-(\sin\left(3x\right)+\sin\left(-x\right))$
The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors
Combine and simplify all terms in the same fraction with common denominator $2$
Simplify the fraction $\frac{2\sin\left(x\right)}{2}$ by $2$
Simplify $\frac{2\sin\left(x\right)}{\sin\left(2x\right)}$
Cancel the fraction's common factor $2$
IV. Check if we arrived at the expression we wanted to prove
Since we have reached the expression of our goal, we have proven the identity