How to Apply Reciprocal Trigonometric Identities?
Reciprocal Trigonometric Identities express each of the six trigonometric ratios as the reciprocal of another ratio. This page covers the reciprocal trigonometric identities formula, product relationships, sign rules, simplification methods and solved examples involving sin, cos, tan, cosec, sec and cot.
On this page
What Are Reciprocal Trigonometric Identities?
The identities can also be written as sin θ = 1/cosec θ, cos θ = 1/sec θ and tan θ = 1/cot θ. Their corresponding products are sin θ × cosec θ = 1, cos θ × sec θ = 1 and tan θ × cot θ = 1, provided the expressions are defined.
Reciprocal Trigonometric Identities Video Lessons
Watch short topic-wise lessons for quick revision.
Pythagorean and Reciprocal Identities
Learn the fundamental Pythagorean and reciprocal identities in trigonometry, including their standard forms and how they relate the six trigonometric ratios.
Reciprocal Trigonometric Identities Formula & Tricks
Important Formulas
Cosecant is the reciprocal of sine.
Secant is the reciprocal of cosine.
Cotangent is the reciprocal of tangent.
Each trigonometric ratio multiplied by its reciprocal equals 1 when both quantities are defined.
These forms show why tan and cot are reciprocal, subject to non-zero denominators.
Quick Tricks
To change a ratio into its reciprocal, interchange the numerator and denominator in its fraction form: sin and cosec, cos and sec, tan and cot.
A ratio multiplied by its reciprocal becomes 1. Pair sin with cosec, cos with sec, or tan with cot before expanding any expression.
A reciprocal is defined only when the original value is non-zero. For example, sec θ = 1/cos θ requires cos θ ≠ 0.
Reciprocal Trigonometric Identities Concepts
The Three Reciprocal Pairs
In right-triangle notation, sin θ = perpendicular/hypotenuse, so cosec θ = hypotenuse/perpendicular. Similarly, cos θ = base/hypotenuse and sec θ = hypotenuse/base; tan θ = perpendicular/base and cot θ = base/perpendicular.
Converting Expressions Using Reciprocals
For example, cosec θ + sec θ becomes 1/sin θ + 1/cos θ. Conversely, 1/sin θ can be written as cosec θ. The replacement must preserve restrictions such as sin θ ≠ 0 or cos θ ≠ 0.
Reciprocal Identities with Pythagorean Identities
The standard related identities are 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ. For example, sec²θ − tan²θ = 1 and cosec²θ − cot²θ = 1.
Solving Equations with Reciprocal Ratios
If cosec θ = 2, then sin θ = 1/2. If sec θ = 5/3, then cos θ = 3/5. The sign of the reciprocal remains the same: the reciprocal of −2 is −1/2.
Reciprocal Trigonometric Identities Data Table
Use the structured data below while following the explanation and examples.
Reciprocal Trigonometric Ratios
Each trigonometric ratio and its reciprocal counterpart.
| Ratio | Reciprocal ratio | Identity |
|---|---|---|
| sin θ | cosec θ | cosec θ = 1/sin θ |
| cos θ | sec θ | sec θ = 1/cos θ |
| tan θ | cot θ | cot θ = 1/tan θ |
Note: The denominator in each reciprocal expression must be non-zero.
Learn more concepts on PrepShots
Start ₹1 Trial →Practice Reciprocal Trigonometric Identities Questions
Practise published questions related to this topic.
Want more practice questions?
Practice in App →Reciprocal Trigonometric Identities Quick Quiz
Attempt 5 questions and check your score instantly.
Keep practising
Practice more Reciprocal Trigonometric Identities questions in the PrepShots app and continue from your current topic.
Practice More Questions - Start ₹1 Trial →Quick Revision Notes
Reciprocal Trigonometric Identities Revision Points
Use these direct rules while simplifying reciprocal trigonometric expressions.
- cosec θ = 1/sin θ and sin θ = 1/cosec θ.
- sec θ = 1/cos θ and cos θ = 1/sec θ.
- cot θ = 1/tan θ and tan θ = 1/cot θ.
- sin θ · cosec θ = cos θ · sec θ = tan θ · cot θ = 1, wherever defined.
- The reciprocal of a negative ratio is also negative.
- Use 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ with reciprocal identities.
- A reciprocal cannot be taken when the original ratio is zero.
Reciprocal Trigonometric Identities FAQs
What is the reciprocal trigonometric identities formula?
The formulas are cosec θ = 1/sin θ, sec θ = 1/cos θ and cot θ = 1/tan θ. Their reverse forms are sin θ = 1/cosec θ, cos θ = 1/sec θ and tan θ = 1/cot θ.
What is the reciprocal of tan θ?
The reciprocal of tan θ is cot θ because cot θ = 1/tan θ. For example, if tan θ = 2/3, then cot θ = 3/2.
If cosec θ = 5/2, what is sin θ?
sin θ = 1/cosec θ = 1/(5/2) = 2/5.
If cos θ = −3/7, what is sec θ?
sec θ = 1/cos θ = 1/(−3/7) = −7/3.
What is the value of sec θ × cos θ?
It is 1, because sec θ = 1/cos θ. This result applies where cos θ is non-zero.
Are reciprocal identities and quotient identities the same?
No. Reciprocal identities pair sin with cosec, cos with sec and tan with cot. Quotient identities are tan θ = sin θ/cos θ and cot θ = cos θ/sin θ.