Pythagorean Trigonometric Identities with Solved Examples

Pythagorean Trigonometric Identities are equations connecting the six trigonometric ratios. They are derived from the Pythagorean theorem and help find unknown ratios, simplify expressions and solve trigonometric questions. The three basic identities are sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.

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What Are Pythagorean Trigonometric Identities?

Pythagorean Trigonometric Identities are trigonometric equations derived from the right-triangle relation a² + b² = c². They connect squares of trigonometric ratios for the same angle θ.

The fundamental identity is sin²θ + cos²θ = 1. Dividing it by cos²θ gives 1 + tan²θ = sec²θ, and dividing it by sin²θ gives 1 + cot²θ = cosec²θ. For example, if sin θ = 3/5 and θ is acute, then cos θ = 4/5 because (3/5)² + (4/5)² = 1.

Pythagorean Trigonometric Identities Video Lessons

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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.

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Pythagorean Trigonometric Identities Formula & Tricks

Important Formulas

Fundamental identity
sin²θ + cos²θ = 1

This identity follows by dividing the Pythagorean theorem by the square of the hypotenuse.

Tangent-sec² identity
1 + tan²θ = sec²θ

Divide sin²θ + cos²θ = 1 by cos²θ, where cos θ ≠ 0.

Cotangent-cosec² identity
1 + cot²θ = cosec²θ

Divide sin²θ + cos²θ = 1 by sin²θ, where sin θ ≠ 0.

Useful rearrangements
cos²θ = 1 − sin²θ; sin²θ = 1 − cos²θ; tan²θ = sec²θ − 1; cot²θ = cosec²θ − 1

Rearrange the basic identities to find a missing squared ratio.

Quick Tricks

Choose the identity from the given ratios

Use sin and cos together with sin²θ + cos²θ = 1. Use tan and sec together with sec²θ − tan²θ = 1. Use cot and cosec together with cosec²θ − cot²θ = 1.

Example: sec²θ − tan²θ = 1, so sec²θ = 169/144 when tan θ = 5/12 and θ is acute.
Use the Pythagorean triple directly

For a 3-4-5 right triangle, the ratios are sin θ = 3/5, cos θ = 4/5, tan θ = 3/4, sec θ = 5/4, cosec θ = 5/3 and cot θ = 4/3, depending on which leg is opposite θ.

Example: sin²θ + cos²θ = (3/5)² + (4/5)² = 9/25 + 16/25 = 1.
Check the quadrant before selecting a sign

An identity determines a squared ratio, but not always its sign. Use the quadrant or the stated range of θ to choose the positive or negative value.

Example: If sin θ = −3/5 and θ lies in the fourth quadrant, then cos θ = 4/5, not −4/5.

Pythagorean Trigonometric Identities Concepts

Fundamental Sine-Cosine Identity

The fundamental Pythagorean identity is sin²θ + cos²θ = 1.

In a right triangle, let the opposite, adjacent and hypotenuse sides be p, b and h. Since p² + b² = h², dividing by h² gives (p/h)² + (b/h)² = 1, which is sin²θ + cos²θ = 1. The notation sin²θ means (sin θ)².

Example: If cos θ = 12/13, then sin²θ = 1 − (12/13)² = 25/169. For an acute angle, sin θ = 5/13.

Tangent and Secant Identity

The tangent-secant identity is 1 + tan²θ = sec²θ.

Dividing sin²θ + cos²θ = 1 by cos²θ gives tan²θ + 1 = sec²θ because tan θ = sin θ/cos θ and sec θ = 1/cos θ. This form applies where cos θ is non-zero.

Example: If tan θ = 5/12 and θ is acute, then sec²θ = 1 + 25/144 = 169/144, so sec θ = 13/12.

Cotangent and Cosecant Identity

The cotangent-cosecant identity is 1 + cot²θ = cosec²θ.

Dividing sin²θ + cos²θ = 1 by sin²θ gives 1 + cot²θ = cosec²θ because cot θ = cos θ/sin θ and cosec θ = 1/sin θ. This form applies where sin θ is non-zero.

Example: If cot θ = 7/24 and θ is acute, then cosec²θ = 1 + 49/576 = 625/576, so cosec θ = 25/24.

Finding Signs in Different Quadrants

Pythagorean identities give squared values, while the quadrant determines the sign of the ratio.

For an acute angle, all six standard ratios are positive. In the second quadrant, sine and cosecant are positive; in the third quadrant, tangent and cotangent are positive; in the fourth quadrant, cosine and secant are positive. The remaining ratios in each quadrant are negative.

Example: If sin θ = −3/5 and θ is in the fourth quadrant, then cos²θ = 1 − 9/25 = 16/25. Therefore cos θ = 4/5 and tan θ = −3/4.

Pythagorean Trigonometric Identities Visual Guide

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3-4-5 Right Triangle for Pythagorean Identity

Visual guide: 3-4-5 Right Triangle for Pythagorean Identity
Opposite side: 3 Adjacent side: 4 Hypotenuse: 5 Angle: θ
For angle θ, the opposite side is 3, the adjacent side is 4 and the hypotenuse is 5; hence sin²θ + cos²θ = (3/5)² + (4/5)² = 1.

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Practice Pythagorean Trigonometric Identities Questions

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1The price of petrol is increased by 28%. A person wants to increase his expenditure by 22% only. By approximately what percent should he decrease his consumption?→ 2An article is sold for Rs. 288 after successive discounts of 20% and 25%. What is the marked price of the article?→ 321.6 ÷ 3.6 × 2 + 0.25 × 16 ÷ 4 - 6 is equal to∶→ 4What is the value of x so that the seven digit number 55350x2 is divisible by 72?→ 5Two articles are sold for Rs. 4,752 each. On one, the seller gains 32% and on the other he loses 28%. What is his overall gain or loss percentage, correct to one decimal place?→ 6The price of sugar is increased by 17%. A person wants to increase his expenditure by 5% only. By approximately what percent should he decrease his consumption?→ 7The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:→ 8What is the value of x so that the seven digit number 8439x53 is divisible by 99?→ 9Two articles are sold for Rs. 962 each. On one, the seller gains 30% and on the other he loses 26%. What is his overall gain or loss percentage, nearest to one decimal place?→ 10An article is sold for Rs. 288 after successive discounts of 25% and x%. If the marked price of the article is Rs. 480, what is the value of x?→

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Quick Revision Notes

Pythagorean Trigonometric Identities Revision Points

Use these identities to calculate missing ratios and simplify trigonometric expressions.

  • sin²θ + cos²θ = 1
  • 1 + tan²θ = sec²θ
  • 1 + cot²θ = cosec²θ
  • sec²θ − tan²θ = 1 and cosec²θ − cot²θ = 1
  • tan θ = sin θ/cos θ and cot θ = cos θ/sin θ
  • sec θ = 1/cos θ and cosec θ = 1/sin θ
  • Use the quadrant to select the sign after finding a squared ratio.
  • The identities are valid only where the involved denominators are non-zero.

Related Topic Explainers

SOH CAH TOA → Trigonometric Ratios of Complementary Angles →

Pythagorean Trigonometric Identities FAQs

What is the main Pythagorean trigonometric identity?

The main identity is sin²θ + cos²θ = 1.

How is 1 + tan²θ = sec²θ obtained?

Divide sin²θ + cos²θ = 1 by cos²θ. This gives tan²θ + 1 = sec²θ, provided cos θ ≠ 0.

How is 1 + cot²θ = cosec²θ obtained?

Divide sin²θ + cos²θ = 1 by sin²θ. This gives 1 + cot²θ = cosec²θ, provided sin θ ≠ 0.

If sin θ = 5/13 and θ is acute, what is cos θ?

cos²θ = 1 − 25/169 = 144/169. Since θ is acute, cos θ = 12/13.

If tan θ = 3/4, what is sec θ for an acute angle?

sec²θ = 1 + (3/4)² = 25/16. Therefore sec θ = 5/4 for an acute angle.

What is the value of sec²θ − tan²θ?

The value is 1, because 1 + tan²θ = sec²θ.

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