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Trigonometric identitiesTrigonometryHigh schoolabout 1:13

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

A narrated, step-by-step trigonometric identities video lesson with captions. Final answer: The identity is true (for every value where both sides are defined)..

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
    sec⁡(x)−cos⁡(x)=sin⁡(x)tan⁡(x)\sec\left( x\right)-\cos\left( x\right) = \sin\left( x\right) \tan\left( x\right)
  2. 2. Steps, part 1
    1. sec⁡θ=1cos⁡θ,tan⁡θ=sin⁡θcos⁡θ\sec\theta = \frac{1}{\cos\theta},\quad \tan\theta = \frac{\sin\theta}{\cos\theta}
      Rewrite everything in terms of sin x and cos x
    2. sin⁡2xcos⁡x\frac{\sin^{2} x}{\cos x}
      Left side in sine and cosine
    3. sin⁡2xcos⁡x\frac{\sin^{2} x}{\cos x}
      Right side in sine and cosine
    4. LHS−RHS=ND,N=−cos⁡2x−sin⁡2x+1\text{LHS} - \text{RHS} = \frac{N}{D},\quad N = -\cos^{2} x - \sin^{2} x + 1
      Subtract the sides and write the difference over a common denominator
  3. 3. Steps, part 2
    1. N=0N = 0
      Replace every cos²x with 1 − sin²x and collect terms
    2. The numerator is identically zero
  4. 4. The answer
    sec⁡(x)−cos⁡(x)≡sin⁡(x)tan⁡(x)\sec\left( x\right)-\cos\left( x\right) \equiv \sin\left( x\right) \tan\left( x\right)

    LHS − RHS reduces to 0

  5. 5. Check
    • Numeric spot check at 10 sample values: both sides agree.

Transcript

The problem. Here is the problem: sec(x) minus cos(x) equals sin(x)tan(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 identically zero. Every term cancelled after using sin squared plus cos squared equals 1, so both sides are equal wherever they are defined.

The answer. So the answer is The identity is true (for every value where both sides are defined). After rewriting both sides with sine and cosine and replacing cos squared with 1 minus sin squared, the difference of the two sides is exactly 0, so sec(x) minus cos(x) equals sin(x)tan(x) holds for every x at which both sides are defined.

Check. Numeric spot check at 10 sample values: both sides agree.

About trigonometric identities

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

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Is sec(x) - cos(x) = sin(x)tan(x) an identity? (Lesson)