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Exercise

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

Step-by-step Solution

1

Combine $1+\frac{\sin\left(x\right)^2}{\cos\left(x\right)^2}$ in a single fraction

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

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

$\frac{\sec\left(x\right)}{\frac{1}{\cos\left(x\right)^2}}$
Why is sin(x)^2 + cos(x)^2 = 1 ?
3

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

$\sec\left(x\right)\cos\left(x\right)^2$
4

Simplify $\sec\left(x\right)\cos\left(x\right)^2$

$\cos\left(x\right)^{1}$
5

Any expression to the power of $1$ is equal to that same expression

$\cos\left(x\right)$

Final answer to the exercise

$\cos\left(x\right)$

Try other ways to solve this exercise

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  • Find the derivative
  • Integrate using basic integrals
  • Verify if true (using algebra)
  • Verify if true (using arithmetic)
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
  • Integrate by parts
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Similar Questions Solved

$\left(\sin\left(x\right)+\cos\left(x\right)\right)^2$

$1+cot^2x$

$tan\left(x\right)csc\left(x\right)$

$\tan\left(x\right)\csc\left(x\right)$

$\frac{2+\tan^2\left(x\right)}{\sec^2\left(x\right)}-1$

$csc^2x-1$

$\frac{2+tan^2x}{sec^2x}-1$

Main Topic: Simplify Trigonometric Expressions

Simplification of trigonometric expressions consists of rewriting an expression with trigonometric functions in a simpler form. To perform this task, we usually use the most common trigonometric identities, and some algebra.

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