Trigonometry problems: Proving trigonometric identities
Trigonometry problems and trigonometric expressions can be simple and very complex. Very often, complex trigonometric expressions can be equivalent to less complicated expressions. Process of verifying that two trigonometric expressions are equivalent (no matter what is the value of an angle) is called proving trigonometric identities.
There are several theories we use when we prove trigonometric identity. Very often, one of the steps when proving identities is to change every expression into sine and cosine.
Example 1: Prove the following identity: cosec x · tg x = sec x.
Solution: We shall work on left side.
Finally, we have reached same expression as on the right side, which proves that identity is correct.
When working with identities, unlike equations, replacement and mathematic operations are performed on only one side of identity. While thinking on how to prove identity, you want to observe both side of equation, but the goal is to trove that each individual side can be transformed into other one. Use every available known trigonometric identity.
Example 2: Prove the following identity: (1 – cos2 x)(1 + ctg2 x) = 1.
Solution: Use basic trigonometric identity and its’ alternative form. From sin2 x + cos2 x = 1 we shall get that sin2 x = 1 – cos2 x. Also replace ctg2 x, and then multiply and shorten.
When working with identities in which you have fractions, use your knowledge of algebra transformations to sum fractions.
Example 3: Prove the following identity:
Solution: Sum two fractions on the left side of the equation by finding common denominator: (1 + cos x) · sin x, and then perform adding.
Now we shall have to apply another algebra technique, square binomial. We also use denominator in factor form, because this way you will be able to shorten something by the end.
Using previous idea, replace sin2 x + cos2 x = 1 and simplify.
Trigonometry problems are easy to learn. If it is possible, factorize trigonometric expressions. Actually, we have already used this in previous example, when we wrote 2 + 2 cos x as 2(1 + cos x), and in this situations, factors have shorten.
Example 4: Prove the following identity:
Solution: Here we have used connection between co-tangent and tangent – ctg x = 1 / tgx.
Now solve dual fraction and shorten.
Finally, we can conclude that, in order to prove trigonometric identity, you can use following ideas:
- Start from the more complicated side of the equation
- Try to turn everything into one trigonometric function, and if this is not possible, tur everything into sine and cosine
- If you encounter fractions, calculate them with application of algebra
- Shorten everything you can, and factor if necessary
- Your goal is to make each side identical to each other. To achieve this, work on one side of the equation, while looking on the other side to get the idea about your next step. If you get stuck, you can try to do something on the other side. This will help because operations are opposite. Trigonometry problems are easy if we are good little technical.