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6.3PureStretch

Using Sec, Cosec and Cot

Now you can simplify expressions, prove identities and solve equations involving the reciprocal functions. The key is to rewrite everything in terms of sin and cos when you get stuck.

28 min Trigonometric Functions
Edexcel A level Maths: 6.3 Using Sec, Cosec and CotCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Simplify expressions with sec, cosec, cot
  • Solve equations involving them
  • Prove identities
  • Convert to sin/cos as a fallback strategy
1

Solving equations

Rewrite a reciprocal equation in terms of cos⁡\cos, sin⁡\sin or tan⁡\tan, then solve as usual. For example sec⁡θ=2\sec\theta = 2 means cos⁡θ=12\cos\theta = \tfrac12.

Example 1Solve cosec⁡θ=2\operatorname{cosec}\theta = 2 for 0≤θ<2π0 \le \theta < 2\pi.
1sin⁡θ=12\sin\theta = \tfrac12.
2θ=π6,5π6\theta = \tfrac{\pi}{6}, \tfrac{5\pi}{6}.
Answer
θ=π6,5π6\theta = \tfrac{\pi}{6}, \tfrac{5\pi}{6}

Tip — Flip the reciprocal first: cosec θ = k ⇒ sin θ = 1/k.

2

Simplifying and proving

When an expression resists simplification, write every term as sin⁡\sin and cos⁡\cos, combine over a common denominator, and use sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1.

Example 1Simplify sin⁡θcot⁡θ\sin\theta \cot\theta.
1sin⁡θ⋅cos⁡θsin⁡θ\sin\theta \cdot \tfrac{\cos\theta}{\sin\theta}.
2=cos⁡θ= \cos\theta.
Answer
cos⁡θ\cos\theta

Formula recap

sec⁡θ=k⇒cos⁡θ=1k\sec\theta = k \Rightarrow \cos\theta = \tfrac1k
Flip to solve.
sin⁡θcot⁡θ=cos⁡θ\sin\theta\cot\theta = \cos\theta
Typical simplification.

Common mistakes to avoid

Trying to “arcsec” directly on a calculator.
Convert to cos/sin/tan first, then use the standard inverse.
Forgetting all solutions in the interval.
Use symmetry/period after flipping the reciprocal.

Key takeaways

  • Flip the reciprocal: cosec θ = k ⇒ sin θ = 1/k.
  • Rewrite stubborn expressions in sin and cos.
  • Use sin²θ + cos²θ = 1 to finish proofs.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. sec⁡θ=2\sec\theta = 2 is equivalent to:

    • cos⁡θ=12\cos\theta = \tfrac12
    • sin⁡θ=12\sin\theta = \tfrac12
    • cos⁡θ=2\cos\theta = 2
    • tan⁡θ=2\tan\theta = 2
    Show answer and explanation

    Answer: cos⁡θ=12\cos\theta = \tfrac12

    Flip.

  2. sin⁡θcot⁡θ\sin\theta\cot\theta simplifies to:

    • cos⁡θ\cos\theta
    • sin⁡θ\sin\theta
    • tan⁡θ\tan\theta
    • 11
    Show answer and explanation

    Answer: cos⁡θ\cos\theta

    cot = cos/sin.

  3. Solutions of cosec⁡θ=2\operatorname{cosec}\theta = 2 on [0,2π)[0, 2\pi):

    • π6,5π6\tfrac{\pi}{6}, \tfrac{5\pi}{6}
    • π3,2π3\tfrac{\pi}{3}, \tfrac{2\pi}{3}
    • π6\tfrac{\pi}{6}
    • π6,7π6\tfrac{\pi}{6}, \tfrac{7\pi}{6}
    Show answer and explanation

    Answer: π6,5π6\tfrac{\pi}{6}, \tfrac{5\pi}{6}

    sin θ = 1/2.

  4. A good fallback when stuck on a trig proof is to:

    • write all in sin and cos
    • differentiate
    • square everything
    • use degrees
    Show answer and explanation

    Answer: write all in sin and cos

    Reduces to known identities.

  5. cos⁡θcosec⁡θ\cos\theta\operatorname{cosec}\theta equals:

    • cot⁡θ\cot\theta
    • tan⁡θ\tan\theta
    • sec⁡θ\sec\theta
    • 11
    Show answer and explanation

    Answer: cot⁡θ\cot\theta

    cos·(1/sin) = cot.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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