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Exercise

$sin\left(3x\right)sin\left(5x\right)+cos\left(3x\right)cos\left(5x\right)$

Step-by-step Solution

1

Using the cosine of a sum formula: $\cos(\alpha\pm\beta)=\cos(\alpha)\cos(\beta)\mp\sin(\alpha)\sin(\beta)$, where angle $\alpha$ equals $3x$, and angle $\beta$ equals $5x$

$\cos\left(3x-5x\right)$
2

Combining like terms $3x$ and $-5x$

$\cos\left(-2x\right)$
3

Use the odd-even identity $\cos(-\theta)=\cos(\theta)$

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

Final answer to the exercise

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

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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