Exercise 10.2 – Further Applications of Basic Identities | First Year Mathematics 2025
Introduction
Mathematics is a subject of logic, reasoning, and problem-solving. In first-year mathematics (FSc Part 1 or equivalent), one of the most important topics is Trigonometric Identities. In the New First Year Math 2025 Edition, Exercise 10.2 is based on Further Applications of Basic Identities.
This exercise teaches you how to apply the Fundamental Law of Trigonometry and other basic formulas to simplify expressions, prove identities, and solve trigonometric problems. It is highly important for short questions, long questions, and MCQs in board exams.
Fundamental Law of Trigonometry
The most important formula in this exercise is:
sin²θ + cos²θ = 1
From this, we can also write:
cos²θ = 1 – sin²θ
sin²θ = 1 – cos²θ
This law works for every angle θ and is the base for many other formulas in trigonometry. You will use it again and again while solving questions in this exercise.
Important Formulas to Remember
Before starting Exercise 10.2, make sure you know these by heart:
1. Reciprocal Formulas
csc θ = 1 / sin θ
sec θ = 1 / cos θ
cot θ = 1 / tan θ
2. Quotient Formulas
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ
3. Pythagorean Formulas
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
4. Negative Angle Formulas
sin(-θ) = -sin θ
cos(-θ) = cos θ
tan(-θ) = -tan θ
These formulas will help you in almost every question of Exercise 10.2.
Types of Questions in Exercise 10.2
In this exercise, you will mostly solve:
1. Proving Identities – You get LHS (Left Hand Side) and RHS (Right Hand Side) and prove both are equal.
2. Simplifying Expressions – Reduce a complicated trigonometric expression into a simpler one.
3.Multiple Formula Applications – Use 2 or more formulas together to reach the answer.
4. Reducing Higher Powers – Use sin²θ + cos²θ = 1 to replace sin²θ or cos²θ.
Step-by-Step Method to Solve
1. Look at the question carefully and decide which side (LHS or RHS) is more complicated.
2. Start working on the complicated side.
3. Change sec, csc, tan, and cot into sin and cos if possible.
4. Apply sin²θ + cos²θ = 1 whenever you see sin²θ or cos²θ.
5. Use reciprocal and quotient formulas to simplify.
6. Factorize if possible and cancel common terms.
7. Continue step-by-step until you get the other side.
Example Question
Prove:
(sin²θ / cos²θ) + (cos²θ / sin²θ) = sec²θ + csc²θ – 2
Solution:
1. Start with LHS:
(sin²θ / cos²θ) + (cos²θ / sin²θ)
2. Take LCM:
(sin⁴θ + cos⁴θ) / (sin²θ cos²θ)
3. Formula: sin⁴θ + cos⁴θ = (sin²θ + cos²θ)² – 2 sin²θ cos²θ
4. Since sin²θ + cos²θ = 1:
(1 – 2 sin²θ cos²θ) / (sin²θ cos²θ)
5. Split into two parts:
1 / (sin²θ cos²θ) – 2
6. Write as:
(sec²θ csc²θ) – 2
7. sec²θ csc²θ = sec²θ + csc²θ – 2
8. Therefore proved.
PDF SOLUTION:
Why This Exercise is Important for Board Exams
Past Paper Repetition – Many of these questions appear every year.
Foundation for Higher Classes – You will need these skills for advanced trigonometry.
MCQs and Short Questions – Quick application of formulas is common in objective papers.
Common Mistakes Students Make
Skipping small steps and getting confused.
Forgetting negative sign rules.
Mixing up formulas like 1 + tan²θ = sec²θ with 1 + cot²θ = csc²θ.
Trying to start from both LHS and RHS in proofs (not recommended unless allowed).
Tips for Preparation
Memorize all formulas daily.
Practice without looking at the answer.
Write each step clearly.
Time yourself while solving to improve speed.
Conclusion
Exercise 10.2 is not just a practice exercise — it’s a core skill-builder for trigonometry. By mastering the Fundamental Law of Trigonometry, reciprocal formulas, quotient formulas, and Pythagorean formulas, you will be able to solve proofs and simplifications confidently.
Regular practice will help you in board exams, in MCQs, and in advanced mathematics topics in future classes.
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