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Reciprocal trig ratios

Define secant, cosecant and cotangent; evaluate them for standard angles using exact values; apply reciprocal identities to simplify trigonometric expressions; prove simple identities involving reciprocal ratios.

Worked examples

Evaluating reciprocal trig ratios for 45°

Straightforward

Problem

Find the exact values of sec⁡45°\sec 45°, csc⁡45°\csc 45° and cot⁡45°\cot 45°.

Simplifying an expression using reciprocal identities

Moderate

Problem

Simplify sec⁡θcsc⁡θ\dfrac{\sec\theta}{\csc\theta}, expressing your answer as a single standard trigonometric ratio.

Proving an identity involving reciprocal ratios

Challenging

Problem

Prove the identity sec⁡2θ−tan⁡2θ=1\sec^2\theta - \tan^2\theta = 1.

Practise

Q1·Straightforward
Find the exact value of csc⁡30°\csc 30°.
Q2·Straightforward
Find the exact value of cot⁡45°\cot 45°.
Q3·Straightforward
Find the exact value of sec⁡60°\sec 60°.
Q4·Straightforward
Which of the following is the exact value of sec⁡30°\sec 30°?
Q5·Moderate
Simplify sec⁡θtan⁡θ\dfrac{\sec\theta}{\tan\theta} using the reciprocal identities.
Q6·Moderate
Simplify cot⁡θcsc⁡θ\dfrac{\cot\theta}{\csc\theta} using the reciprocal identities.
Q7·Moderate
Evaluate sin⁡θ⋅csc⁡θ+cos⁡θ⋅sec⁡θ\sin\theta \cdot \csc\theta + \cos\theta \cdot \sec\theta for any value of θ\theta where both expressions are defined.
Q8·Moderate
Simplify sin⁡2θ⋅sec⁡θtan⁡θ\dfrac{\sin^2\theta \cdot \sec\theta}{\tan\theta} using the reciprocal identities.
Q9·Challenging
Prove the identity sec⁡2θ−tan⁡2θ=1\sec^2\theta - \tan^2\theta = 1 by expressing each term in terms of sin⁡θ\sin\theta and cos⁡θ\cos\theta. What is the value of sec⁡2θ−tan⁡2θ\sec^2\theta - \tan^2\theta?
Q10·Challenging
Simplify sec⁡2θtan⁡θ+cot⁡θ\dfrac{\sec^2\theta}{\tan\theta + \cot\theta} by expressing all ratios in terms of sin⁡θ\sin\theta and cos⁡θ\cos\theta.
Q11·Challenging
Expand and simplify (sec⁡θ−1)(sec⁡θ+1)(\sec\theta - 1)(\sec\theta + 1), expressing your answer as a single trigonometric ratio.
Q12·Challenging
Simplify csc⁡θcot⁡θ+tan⁡θ\dfrac{\csc\theta}{\cot\theta + \tan\theta} by expressing all terms in terms of sin⁡θ\sin\theta and cos⁡θ\cos\theta.