Solved Example on Trigonometric Identities
Example 1: Prove that (1 – sin2θ) sec2θ = 1
Solution:
We have:
LHS = (1 – sin2θ) sec2θ
= cos2θ . sec2θ
= cos2θ . (1/cos2θ)
=1
= RHS.
∴ LHS = RHS. [Hence Proved]
Example 2: Prove that (1 + tan2θ) cos2θ = 1
Solution:
We have:
LHS = (1 + tan2θ)cos2θ
⇒ LHS = sec2θ . cos2θ
⇒ LHS = (1/cos2θ) . cos2θ
⇒ LHS = 1 = RHS.
∴ LHS=RHS. [Hence Proved]
Example 3: Prove that (cosec2θ – 1) tan²θ = 1
Solution:
We have:
LHS = (cosec²θ – 1) tan2θ
⇒ LHS = (1 + cot2θ – 1) tan2θ
⇒ LHS = cot2θ . tan2θ
⇒ LHS = (1/tan2θ) . tan2θ
⇒ LHS = 1 = RHS.
∴ LHS=RHS. [Hence Proved]
Example 4: Prove that (sec4θ – sec2θ) = (tan2θ + tan4θ)
Solution:
We have:
LHS = (sec4θ – sec2θ)
⇒ LHS = sec2θ(sec2θ – 1)
⇒ LHS = (1 + tan2θ) (1 + tan2θ – 1)
⇒ LHS = (1 + tan2θ) tan2θ
⇒ LHS = (tan2θ + tan4θ) = RHS
∴ LHS = RHS. [Hence Proved]
Example 5: Prove that √(sec2θ + cosec2θ) = (tanθ + cotθ)
Solution:
We have:
LHS = √(sec2θ + cosec2θ ) = √((1 + tan2θ) + (1 + cot2θ))
⇒ LHS = √(tan2θ + cot2θ + 2)
⇒ LHS = √(tan2θ + cot2θ + 2tanθ.cotθ ) (tanθ . cotθ = 1)
⇒ LHS = √(tanθ + cotθ)2
⇒ LHS = tanθ + cotθ = RHS
∴ LHS = RHS [Hence Proved]
Trigonometric Identities
Trigonometric Identities are various identities that are used to simplify various complex equations involving trigonometric functions. Trigonometry is a branch of Mathematics that deals with the relationship between the sides and angles of a triangle., These relationships are defined in the form of six ratios which are called trigonometric ratios – sin, cos, tan, cot, sec, and cosec.
In an extended way, the study is also of the angles forming the elements of a triangle. Logically, a discussion of the properties of a triangle; solving a triangle, and physical problems in the area of heights and distances using the properties of a triangle – all constitute a part of the study. It also provides a method of solution to trigonometric equations.
Table of Content
- What are Trigonometric Identities?
- List of Trigonometric Identities
- Reciprocal Trigonometric Identities
- Pythagorean Trigonometric Identities
- Trigonometric Ratio Identities
- Trigonometric Identities of Opposite Angles
- Complementary Angles Identities
- Supplementary Angles Identities
- Periodicity of Trigonometric Function
- Sum and Difference Identities
- Double Angle Identities
- Half Angle Formulas
- Some more Half Angle Identities
- Product-Sum Identities
- Products Identities
- Triple Angle Formulas
- Proof of the Trigonometric Identities
- Relation between Angles and Sides of Triangle
- FAQs on Trigonometric Identities