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Trigonometric identitiesTrigonometryHigh schoolabout 1:46

Is tan(x) = sin(x)cos(x) an identity?: animated lesson

A narrated, step-by-step trigonometric identities video lesson with captions. Final answer: This is not an identity: the two sides are not equal for all values..

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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 (0.7 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
$\tan\left( x\right) = \sin\left( x\right) \cos\left( x\right)$
2. Steps, part 1
1. $\tan\theta = \frac{\sin\theta}{\cos\theta}$
  Rewrite everything in terms of sin x and cos x
2. $\frac{\sin x}{\cos x}$
  Left side in sine and cosine
3. $\sin x\cos x$
  Right side in sine and cosine
4. $\text{LHS} - \text{RHS} = \frac{N}{D},\quad N = -\sin x\cos^{2} x + \sin x$
  Subtract the sides and write the difference over a common denominator
3. Steps, part 2
1. $N = \sin^{3} x$
  Replace every cos²x with 1 − sin²x and collect terms
2. The numerator is not zero, so the sides are not identical
4. The answer
$\tan\left( x\right) \not\equiv \sin\left( x\right) \cos\left( x\right)$
LHS − RHS numerator = sin^3(x)
5. Check
- At x = 30°, the left side ≈ 0.5773502692 but the right side ≈ 0.4330127019.

Transcript

The problem. Here is the problem: tan(x) equals sin(x)cos(x). Prove a trigonometric identity.

Steps, part 1. Rewrite everything in terms of sin x and cos x. Identities used:. Left side in sine and cosine. Right side in sine and cosine. Subtract the sides and write the difference over a common denominator. The identity is true exactly when this numerator is zero for every value of the variable.

Steps, part 2. Replace every cos squaredx with 1 minus sin squaredx and collect terms. The numerator is not zero, so the sides are not identical. At x equals 30°, the left side approximately 0.5773502692 but the right side approximately 0.4330127019.

The answer. So the answer is This is not an identity: the two sides are not equal for all values. The difference of the sides does not reduce to 0. At x equals 30°, the left side approximately 0.5773502692 but the right side approximately 0.4330127019. The equation may still be true for particular values; solve it as an equation to find them.

Check. At x equals 30°, the left side approximately 0.5773502692 but the right side approximately 0.4330127019.

About trigonometric identities

Decide whether a trigonometric equation is an identity and show why.

Is tan(x) = sin(x)cos(x) an identity?: Step-by-Step Lesson