Techniques to Solve Trigonometry and Calculus Problems in UPSC Exam

Trigonometry:
1. Understand the Basics: Ensure you have a strong foundation in trigonometric concepts such as angles, trigonometric functions, identities, and properties.
2. Memorize Key Formulas: Memorize important trigonometric formulas like Pythagorean identities, angle sum and difference identities, double angle identities, and product-to-sum identities.
3. Use Trigonometric Identities: Apply trigonometric identities to simplify expressions and solve equations. Identify which identities are applicable to a given problem.
4. Visualize Problems: Use diagrams to visualize trigonometric relationships and geometric interpretations of trigonometric functions. This can help in understanding and solving problems more effectively.
5. Practice with Previous Year Questions: Solve previous year UPSC questions to get familiar with the type of trigonometry problems asked in the exam. This will help you understand the exam pattern and identify areas to focus on.

Calculus:
1. Master the Fundamentals: Have a strong understanding of basic calculus concepts like limits, derivatives, and integrals. Familiarize yourself with the rules and properties associated with these concepts.
2. Know Differentiation and Integration Techniques: Learn various techniques of differentiation and integration such as product rule, chain rule, quotient rule, integration by substitution, integration by parts, and partial fractions. Practice applying these techniques to solve problems efficiently.
3. Simplify Expressions: Simplify complex expressions using algebraic manipulation before differentiating or integrating. This can help in reducing the complexity of the problem and making it easier to solve.
4. Use Graphs: Utilize graphical representations of functions to understand their behavior, identify critical points, and analyze the nature of functions. Graphs can provide valuable insights for solving calculus problems.
5. Practice Calculus Problems: Solve a wide range of calculus problems, including both theoretical and application-based questions, to improve your problem-solving skills. Practice solving questions from previous year UPSC papers to get acquainted with the exam format and level of difficulty.

Remember, regular practice and thorough understanding of concepts are key to mastering trigonometry and calculus for the UPSC exam.