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Exercise

$\frac{\sin\left(5x\right)+\sin^2x}{\sin\left(5x\right)+\sin\left(x\right)}$

Step-by-step Solution

1

Apply the trigonometric identity: $\sin\left(a\right)+\sin\left(b\right)$$=2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)$, where $a=5x$ and $b=x$

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

Combining like terms $5x$ and $x$

$\frac{\sin\left(5x\right)+\sin\left(x\right)^2}{2\sin\left(\frac{6x}{2}\right)\cos\left(\frac{5x-x}{2}\right)}$
3

Combining like terms $5x$ and $-x$

$\frac{\sin\left(5x\right)+\sin\left(x\right)^2}{2\sin\left(\frac{6x}{2}\right)\cos\left(\frac{4x}{2}\right)}$
4

Take $\frac{6}{2}$ out of the fraction

$\frac{\sin\left(5x\right)+\sin\left(x\right)^2}{2\sin\left(3x\right)\cos\left(\frac{4x}{2}\right)}$
5

Take $\frac{4}{2}$ out of the fraction

$\frac{\sin\left(5x\right)+\sin\left(x\right)^2}{2\sin\left(3x\right)\cos\left(2x\right)}$

Final answer to the exercise

$\frac{\sin\left(5x\right)+\sin\left(x\right)^2}{2\sin\left(3x\right)\cos\left(2x\right)}$

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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