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 Video by Zeeshan Zamurred Trigonometric Functions

Edexcel A level Maths: 6.3 Using Sec, Cosec and CotWatch the full walkthrough 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

Solving equations

Rewrite a reciprocal equation in terms of $cos$ , $sin$ or $tan$ , then solve as usual. For example $sec θ = 2$ means $cos θ = \frac{1}{2}$ .

sin θ = \frac{1}{2}

θ = \frac{π}{6}, \frac{5 π}{6}

Answer

θ = \frac{π}{6}, \frac{5 π}{6}

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

Simplifying and proving

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

sin θ \cdot \frac{c o s θ}{s i n θ}

= cos θ

Answer

cos θ

Formula recap

sec θ = k \Rightarrow cos θ = \frac{1}{k}

Flip to solve.

sin θ cot θ = cos θ

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

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