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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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