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The trick to proving trig identities is intuition, which can only be gained through experience. The more basic formulas you have memorized, the faster you will be. The following identities are essential to all your work with trig functions. Make a point of memorizing them.
tan(x) = sin(x)/cos(x)
cot(x) = cos(x)/sin(x)
csc(x) = 1/sin(x)
sec(x) = 1/cos(x)
cot(x) = 1/tan(x)
sin(x) = 1/csc(x)
cos(x) = 1/sec(x)
tan(x) = 1/cot(x)
sin2(x) + cos2(x) = 1
cot2A +1 = csc2A
1+tan2A = sec2A
(For a list of other important identities, see the Trig Cheat Sheet article in this series.)
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How to Solve Them Correctly Every Time
The following seven step process will work every time. It is rather tedious, and can take more time than necessary. As you gain more practice, you can skip or combine these steps when you recognize other identities.
STEP 1: Convert all sec, csc, cot, and tan to sin and cos. Most of this can be done using the quotient and reciprocal identities.
STEP 2: Check all the angles for sums and differences and use the appropriate identities to remove them.
STEP 3: Check for angle multiples and remove them using the appropriate formulas.
STEP 4: Expand any equations you can, combine like terms, and simplify the equations.
STEP 5: Replace cos powers greater than 2 with sin powers using the Pythagorean identities.
STEP 6: Factor numerators and denominators, then cancel any common factors.
STEP 7: Now, both sides should be exactly equal, or obviously equal, and you have proven your identity.
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Example Problem Using the 7 Step Method
Show that cos4(x) - sin4(x) = cos(2x)
STEP 1: Everything is already in sin and cos, so this part is done.
cos4(x) - sin4(x) = cos (2x)
STEP 2: Since there are no sums or difference inside the angles, this part is done.
cos4(x) - sin4(x) = cos (2x)
STEP 3: cos(2x) is a double angle. Use the double angle formula: cos (2x) = cos2(x) - sin2(x), to simplify.
cos4(x) - sin4(x) = cos2(x) - sin2(x)
STEP 4: Here is where your algebra knowledge comes in. In this case, we can see that the left side is a “difference of two squares"
[if you forgot: a2-b2 = (a+b)(a-b)]
Left side: cos4x - sin4x - (cos2(x))2 - (cos2(x))2 = (cos2(x)-sin2(x))(cos2(x)+sin2(x))
Now, our problem looks like this:
(cos2(x)-sin2x))(cos2(x)+sin2(x))= cos2(x) - sin2(x)
The sides are almost the same
STEP 5: There are no powers greater than 2, so we can skip this step
STEP 6: Since cos2(x) - sin2(x) appears on both sides of the equation, we can cancel it.
We are left with: cos2(x) + sin2(x) = 1
STEP 7: Since this is one of the pythagorean identities, we know it is true, and the problem is done.
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Proving Identities Directly
The 7 step method works both sides and meets in the middle, like a V. Some teachers will ask you to prove the identity directly (from one side to the other in a straight line). That is easily done using the work above. Just write down all the left side parts in order first, then the right side parts in backwards order, so it looks like this:
cos4(x) - sin4(x) = (cos2(x)-sin2(x))(cos2(x)+sin2(x)) = (cos2(x)-sin2(x))(1) = cos2(x)-sin2x = cos(2x)
Or write the right hand steps in order first and then the left hand step backwards so it looks like this:
cos(2x) = cos2(x)-sin2(x) = (cos2(x)-sin2(x))(cos2(x)+sin2(x) = cos4(x) - sin4(x)
Even though this is a simple problem, the same steps will work every time no matter the difficulty.
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- Get both sides of the equation in the same functions. You don’t always have to use sin and cos, but its easier to compare when both sides are composed of similar functions
- Make sure all your angles are the same. Using sin(2A) and sinA is difficult, but if you use sin2A = 2sin(x)cos(x), that leaves sin(x) and cos(x), and now all your functions match. The same goes for addition and subtraction: don’t try working with sin(A+B) and sinA. Instead, use sin(A+B) = sin(x)cos(x)+cos(x)sin(x) so that all the angles match.
- 3 main ways to solve: Convert right side to left side [direct right-left], convert left side to right side [direct left-right], or convert both sides to the same function [meet in the middle]
- If you need to add more powers (or remove them), use cos^2(x) + sin^2(x) = 1. You can always multiply by 1 without changing the meaning, so therefore you can always multiply by cos^2(x)+sin^2(x).
- Once you get the hang of it, you will begin to see patterns. For instance, in the example above, you might notice right off that the left side is difference of two squares and do that first. Then, you quickly simplify to cos^2(x) = sin^2(x), which tells you which double angle identity to use.
- If you keep getting stuck on a problem, take a break. Come back with a clean sheet of paper, and start over from the beginning. Often, it helps to change the direction (from left-right to right-left).
- Let someone else read through your work, just to see if they follow it and can give a new perspective. When you stare at the same equations for too long, you’ll likely start to miss things that you would have noticed at the beginning.