Verifying Trig Identities

© 2004 by Karl Hahn
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In precalculus classes you were probably asked to verify the truth of various equations involving trig functions. The trig functions were always of some variable, x, or θ, or some other independent variable. When you verify the equation, you needed to show that it was true for every value of that independent variable. Once you had done that, you had demonstrated that the equation was an identity.

The methods you use to verify trig identities are to use the basic identities we've already covered on the trig identity page. When the identity involves tan(x), sec(x), cot(x), or csc(x), then a smart move is to replace the function with its corresponding quotient of sine and cosine (e.g.,  tan(x) = sin(x)/cos(x)  and  sec(x) = 1/cos(x)).  Where applicable, you also use various methods you learned in algebra, such as putting things over a common denominator, factoring out a common factor, and so on.

Here are some examples of verifications of trig identities with a battle plan described in each case. You are encouraged follow that battle plan and complete each of the examples.

Example 1:


   sin(x) + cos(x)  =

  cot(x) + 1
            
    csc(x)

Battle plan: Expand cot(x) and csc(x) into quotients. Then multiply top and bottom of the right-hand expression by sin(x).

Example 2:


(sec(x) - tan(x))2  =

   1 - sin(x)
             
   1 + sin(x)

Battle plan: On the left, expand sec(x) and tan(x) into quotients. Note that they already have a common denominator. So add the two quotients on the left. On the right, multiply top and bottom by  1 - sin(x).  Apply the difference of squares to the denominator. Then apply a basic trig identity to the denominator.

Example 3:

          1
                    =
   sec(x) - tan(x)

  sec(x) + tan(x)

Battle plan: Expand sec(x) and tan(x) into quotients on both sides. Again the quotients (on both sides) will already have a common denominator. So use that fact to take the indicated sum on both sides. Multiply top and bottom of the left side by cos(x). Then multiply both sides by  1 - sin(x).  This gives you a cancellation on the left and a difference of squares in the numerator on the right. Apply a basic identity to the numerator on the right, and then take the cancellation that is offered as a result.

Example 4:

   1 + sin(x)
              +
     cos(x)
   cos(x)
             =  2 sec(x)
 1 + sin(x)

Battle plan: Expand the right using the quotient for sec(x). Put the two quotients on the left over a common denominator. On the left you will end up with  (1 + sin(x))2 as part of the numerator. Square it out. Observe that after that you can rearrange the numerator to contain a sum of sin2(x) and cos2(x). Apply a basic identity to that. Gather like terms of the resulting numerator and observe that it now has a common factor with the denominator. Take the cancellation, and you'll be done.

Example 5:


   sec(x + y)  =

    sec(x)sec(y)csc(x)csc(y)
                             
  csc(x)csc(y) - sec(x)sec(y)

Battle plan: Expand all the functions into quotients of sines and cosines. On the left, apply the cosine-of-a-sum identity to the denominator. On the right, multiply top and bottom by  cos(x)cos(y)sin(x)sin(y).  Take the cancellations, and you're done.


Here's one done out for you. All you have to do is follow the steps. At the end, replace the csc(x) on the right with 1/sin(x) and the verification is complete.


Here's one done for you with clickable steps.

Prove that

   cos(2θ) - cos(4θ)
                      =  tan(θ)
   sin(2θ) + sin(4θ)

See step by step solution


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