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Trigonometric identitiesTrigonometryHigh schoolabout 1:24

Is sin(x) + cos(x) = 1 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..

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 (0.5 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    sin⁡(x)+cos⁡(x)=1\sin\left( x\right)+\cos\left( x\right) = 1
  2. 2. Steps, part 1
    1. Rewrite everything in terms of sin x and cos x
    2. cos⁡x+sin⁡x\cos x + \sin x
      Left side in sine and cosine
    3. 11
      Right side in sine and cosine
    4. LHS−RHS=ND,N=cos⁡x+sin⁡x−1\text{LHS} - \text{RHS} = \frac{N}{D},\quad N = \cos x + \sin x - 1
      Subtract the sides and write the difference over a common denominator
  3. 3. Steps, part 2
    1. N=cos⁡x+sin⁡x−1N = \cos x + \sin x - 1
      Replace every cos²x with 1 − sin²x and collect terms
    2. The numerator is not zero, so the sides are not identical
  4. 4. The answer
    sin⁡(x)+cos⁡(x)≢1\sin\left( x\right)+\cos\left( x\right) \not\equiv 1

    LHS − RHS numerator = cos(x) + sin(x) - 1

  5. 5. Check
    • At x = 30°, the left side ≈ 1.366025404 but the right side ≈ 1.

Transcript

The problem. Here is the problem: sin(x) plus cos(x) equals 1. Prove a trigonometric identity.

Steps, part 1. Rewrite everything in terms of sin x and cos x. The expression already uses only sine and cosine. 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 1.366025404 but the right side approximately 1.

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 1.366025404 but the right side approximately 1. 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 1.366025404 but the right side approximately 1.

About trigonometric identities

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

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Is sin(x) + cos(x) = 1 an identity?: Step-by-Step Lesson