This is more of an opinion question:
I'm working on Verifying Identities but I always have trouble knowing where to start. Once my teacher points it out i'm like "oh, duh" but on my own I have not the slightest idea on how to begin these; especially the tricky ones. Does anyone have any tips/tricks they use to verify identities??
Two general rules I usually follow are:
1. Start with the more complicated side and try to reduce it to the simpler side
2. Unless I recognize some well-known identities, I try to change everything to sines and cosines.
Verifying identities in mathematics can indeed be challenging, but with some strategies and practice, you can become better at it. Here are some tips and tricks to help you:
1. Start with the more complicated side: When verifying an identity, it's often helpful to start with the more complex side of the equation. Simplify it using algebraic manipulations, applying properties of equality, factoring, expanding, or any other techniques you have learned.
2. Work on both sides of the equation simultaneously: Instead of carrying out simplifications on one side and leaving the other side untouched, it's often beneficial to work on both sides of the equation simultaneously. This helps maintain balance and identify opportunities for canceling or combining terms.
3. Look for common denominators: If you have fractions involved, it can be helpful to find a common denominator and simplify the fractions before proceeding. This can often lead to cancellation of terms and help reveal the underlying identity.
4. Utilize trigonometric identities: Familiarize yourself with common trigonometric identities such as Pythagorean identities, reciprocal identities, double-angle identities, and sum/difference formulas. These identities can be useful for simplifying expressions and establishing equivalences.
5. Substitute known identities: If you encounter complex expressions involving trigonometric functions, it can be beneficial to substitute known trigonometric identities into the equation to simplify it. This can help in transforming complex expressions into simpler ones that are easier to work with.
6. Be mindful of your steps: Pay attention to each step you take while simplifying the equation. Be cautious with algebraic manipulations and avoid making arithmetic or algebraic errors. Keep track of each transformation to ensure you don't lose any information or introduce false assumptions.
7. Practice consistently: Verifying identities can be challenging, but consistent practice is key. Work through a variety of identity problems, starting with simpler ones and gradually moving to more complex ones. Repeat the process regularly, and with time, you'll become more comfortable and adept at identifying patterns and strategies for verifications.
Remember, verifying identities requires practice, patience, and a systematic approach. Don't get discouraged if it takes time to grasp certain concepts or techniques. Keep exploring, seeking guidance from your teacher or classmates when needed, and gradually build your confidence in solving identity problems.