Verifying trigonometric identities Kowalski Identities. An identity is an equation where both the left-hand side and the right-hand side match for every input value in the domain of the equation. To verify an identity means to show that one side of the equation can be rewritten using the rules of algebra and trigonometry into the other side of the equation. The following algorithm is often helpful in verifying identities. 1. Choose a side. Start of with one side of the equation, either the LHS or the RHS. A good rule of thumb is start with the more complicated side. I personally suggest you also include parentheses when appropriate, and write things like sin(x)3 rather than sin3 x. 2. Convert to sines and cosines. We tend to understand sinusoids better, so it’s convenient to work with them. (Getting used to converting into sine and cosines is also a good Calc 1 skill.) 3. Fix up fractions. If fractions are present, it’s a good idea to rewrite the expression as a single, simpliﬁed fraction. Some things to keep in mind: • Always simplify fractions-in-fractions ﬁrst. A useful technique is to enclose the fraction in paren- theses and then multiply the top and bottom of the fraction by whatever it takes to eliminate all the little sub-denominators. • Put things on a common denominator. If there are several terms added together, put them all on a common denominator and add the numerators together. 4. Expand and/or factor. The idea is to look for combinations (which you ﬁnd by expanding) or cancelations (which you ﬁnd by factoring), in the hope of further simplifying the expression. Some things to keep in mind: • Expand. If you have parenthetical expressions, expand them out completely and collect common terms. Common techniques include the distributive law c(a + b) = ca + cb and FOIL (a + b)(c + d) = ac + ad + bc + bd. • Factor. If a trig expression is common to all terms, it should be factored out: ca + cb = c(a + b). Also, keep an eye out for opportunities to use the diﬀerence of squares formula a2 − b2 = (a + b)(a − b). • Angle addition or subtraction. Remember that you cannot distribute trig terms, or factor constants out of them. However, you can use the angle-addition identities in such cases. They are sin(a ± b) = sin(a) cos(b) ± cos(a) sin(b), cos(a ± b) = cos(a) cos(b) sin(a) sin(b). 5. Pythagorean Identity. If you have terms of the form sin(x)2 or cos(x)2 , you might try using the Pythagorean Identity sin(x)2 + cos(x)2 = 1 to rewrite it. Once you’ve rewritten the expression, try Step 4 all over again and see if it works. 6. Switch sides. If you cannot ﬁgure out how to get your current expression to match the other side of the identity, try starting instead with that other side, and repeat the process above. Sometimes the less complicated side is actually easier to work with. If you cannot still obtain the other side, but you can rewrite both the LHS and the RHS into a same common expression, then you’re also done!