Trigonometric Identities – Solved Problems
Trigonometric identities are equations involving trigonometric ratios that are true for all permissible values of the angle. These identities form the foundation of trigonometry and are widely used in algebra, geometry, and calculus.
In this article, we solve common and important trigonometric identity problems step by step using standard identities.
Problem 1: Prove an Identity
Prove that:
(sinθ / cosθ) + (cosθ / sinθ) = 1 / (sinθ cosθ)
Solution:
Start with the Left Hand Side (LHS):
Take LCM of denominators:
Using identity:
This equals the Right Hand Side (RHS).
Hence proved.
Problem 2: Verify an Identity
Verify:
(1 − tan²θ) / (1 + tan²θ) = cos 2θ
Solution:
Using identity:
Substitute:
Hence verified.
Problem 3: Simplify the Expression
Simplify:
(sinθ / (1 + cosθ)) + ((1 + cosθ) / sinθ)
Solution:
Take LCM:
Expand numerator:
Using identity sin²θ + cos²θ = 1:
Cancel common terms:
Answer: 2 cosecθ
Problem 4: Prove the Identity
Prove that:
(1 / cosecθ − secθ) = (sinθ − cosθ) / (sinθ cosθ)
Solution:
Convert into sine and cosine:
Take LCM:
Rewrite RHS:
Both sides simplify to the same form.
Hence proved.
Problem 5: Prove Using Identities
Prove that:
(secθ − tanθ)(secθ + tanθ) = 1
Solution:
Using identity:
Using identity:
Hence proved.
Problem 6: Simplify
Simplify:
(1 − sin²θ) / (1 + tan²θ)
Solution:
Using identities:
Answer: cos⁴θ
Problem 7: Prove the Identity
Prove that:
(1 + tan²θ)(1 − sin²θ) = 1
Solution:
Hence proved.
Conclusion: Trigonometric identities simplify complex expressions and form the backbone of trigonometry. Regular practice of these solved problems helps build speed, accuracy, and conceptual clarity for examinations.