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Simplifying expressionsAlgebraHigh schoolabout 0:48

Simplify tan(x)cos(x): animated lesson

A narrated, step-by-step simplifying expressions video lesson with captions. Final answer: sin(x).

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Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.3 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
$\tan\left( x\right) \cos\left( x\right)$
2. Step by step
1. $\tan\theta = \frac{\sin\theta}{\cos\theta}$
  Rewrite everything in terms of sin x and cos x
2. $\frac{\sin x\cos x}{\cos x}$
  Combine into a single fraction
3. $\sin x$
  Use sin²x + cos²x = 1 (replace cos² with 1 − sin²) and cancel common factors
4. $\sin x = \sin x$
  Recognise a standard form
3. The answer
$\sin x$
tan(x)cos(x) = sin(x)
4. Check
- Numeric spot check: the original expression and the result agree at sample values of x.

Transcript

The problem. Here is the problem: tan(x)cos(x). Simplify a trigonometric expression.

Step by step. Rewrite everything in terms of sin x and cos x. Identities used:. Combine into a single fraction. Use sin squaredx plus cos squaredx equals 1 (replace cos squared with 1 minus sin squared) and cancel common factors. Recognise a standard form. Cross-multiplying and reducing with sin squared plus cos squared equals 1 shows the difference is exactly 0.

The answer. So the answer is sin(x). tan(x)cos(x) simplifies to sin(x) for every x where the original expression is defined.

Check. Numeric spot check: the original expression and the result agree at sample values of x.

About simplifying expressions

Simplify algebraic and rational expressions by collecting like terms and cancelling common factors.

Simplify tan(x)cos(x): Animated Step-by-Step Lesson