Question

tricks for solving trigonometry questions

Answer 1

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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.
Example Problem Using the 7 Step Method

SOOOO SIMPLE

Answer 2

What are the tips to solve trigonometry problems fast?

{Original Reference: }

Guide to Proving Trig Identities

written by: Kayla Griffin • edited by: Noreen Gunnell • updated: 3/26/2014

IT MIGHT TAKE SOME TIME TO READ THIS BUT IT IS WORTH IT!

Proving identities is a big part of any trigonometry class (or method of study). Here, you will find a basic method that will work on every problem, an example of how to use it, and additional tips and tricks to save you some time.

Essential IdentitiesThe 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.Quotient Identities:
How to Solve Them Correctly Every TimeThe 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.
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.