Solve a problem

Trigonometric identitiesTrigonometryHigh schoolabout 1:30

Is cos(2x) = 2cos(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..

SpeedLanguage

Chapters

Transcript

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 (0.6 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
$\cos\left(2x\right) = 2\cos\left( x\right)$
2. Steps, part 1
1. $\sin(2x) = 2\sin x\cos x,\quad \cos(2x) = \cos^{2} x - \sin^{2} x$
  Rewrite everything in terms of sin x and cos x
2. $-2\sin^{2} x + 1$
  Left side in sine and cosine
3. $2\cos x$
  Right side in sine and cosine
4. $\text{LHS} - \text{RHS} = \frac{N}{D},\quad N = \cos^{2} x - \sin^{2} x - 2\cos x$
  Subtract the sides and write the difference over a common denominator
3. Steps, part 2
1. $N = -2\sin^{2} x - 2\cos x + 1$
  Replace every cos²x with 1 − sin²x and collect terms
2. The numerator is not zero, so the sides are not identical
4. The answer
$\cos\left(2x\right) \not\equiv 2\cos\left( x\right)$
LHS − RHS numerator = -2·sin^2(x) - 2·cos(x) + 1
5. Check
- At x = 30°, the left side ≈ 0.5 but the right side ≈ 1.732050808.

Transcript

The problem. Here is the problem: cos(2x) equals 2cos(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 0.5 but the right side approximately 1.732050808.

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 0.5 but the right side approximately 1.732050808. 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 0.5 but the right side approximately 1.732050808.

About trigonometric identities

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

Is cos(2x) = 2cos(x) an identity?: Step-by-Step Lesson