Question

# How to get an idea of how to start solving trigonometry sums

Solution

## 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.

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