Introduction to Trigonometry | Class 10 Mathematics Notes

8.4 Trigonometric Identities

Trigonometric identities are equations involving trigonometric ratios that hold true for all values of the angle(s) within their domain. These identities are fundamental tools for simplifying expressions and solving equations in trigonometry.

The section begins by proving the primary identity using the Pythagoras theorem in a right triangle ABC right angled at B:

AB² + BC² = AC²

Dividing throughout by AC² gives:

(AB/AC)² + (BC/AC)² = 1

Recognizing the ratios, this becomes:

cos² A + sin² A = 1

This identity holds for all angles A where 0° ≤ A ≤ 90°.

Next, dividing the Pythagorean relation by AB² yields:

1 + tan² A = sec² A

This identity holds for 0° ≤ A < 90°, as tan and sec are undefined at 90°.

Similarly, dividing by BC² gives:

cot² A + 1 = cosec² A

valid for 0° < A ≤ 90°.

These identities allow expressing any trigonometric ratio in terms of others. For example, if tan A = 1/√3, then cot A = √3, sec² A = 1 + tan² A = 4/3, so sec A = 2/√3, cos A = √3/2, sin A = 1/2, and csc A = 2.

The section also includes examples proving identities such as:

• Expressing cos A, tan A, and sec A in terms of sin A.
• Proving sec A(1 - sin A)(sec A + tan A) = 1.
• Proving (cot A - cos A)/(cot A + cos A) = (cosec A - 1)/(cosec A + 1).
• Proving (sin θ - cos θ + 1)/(sin θ + cos θ - 1) = 1/(sec θ - tan θ).

These proofs involve algebraic manipulation, substitution of trigonometric ratios, and application of the fundamental identities.

Mastering these identities is crucial for simplifying complex trigonometric expressions and solving equations efficiently.

📊 Diagram: Fig. 8.21 shows right triangle ABC right angled at B used to derive the Pythagorean identity by relating sides AB, BC, and AC.

🧪 Activity: Students are guided to prove identities using algebraic manipulation and substitution of trigonometric ratios.

🔗 Connection: Summarizes the chapter and prepares for application of these identities in advanced problems.