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1/1-cosx+1/1+cosx=2csc^2x

Prove the Trigonometic identity

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Tamara J. | Math Tutoring - Algebra and Calculus (all levels)Math Tutoring - Algebra and Calculus (al...
4.9 4.9 (51 lesson ratings) (51)
2

You are looking to prove the following:     1/(1-cosx) + 1/(1+cosx) = 2csc2x

To prove that the left hand side of the equation does in fact equal the right hand side of the equation, we start by solving for the left hand side of the equation.

     1/(1-cosx)  +  1/(1+cosx)

Since we are essentially adding two fractions, we need to make it so that both fractions have a common denominator. To do so, we multiply the first term by:  (1+cosx)/(1+cosx), and the second term by:  (1-cosx)/(1-cosx). Notice that  (1+cosx)/(1+cosx) = 1   and that  (1-cosx)/(1-cosx) = 1, so you are not actually altering the value of the terms you are multiplying them by.

     [1/(1-cosx) * (1+cosx)/(1+cosx)] + [1/(1+cosx) * (1-cosx)/(1-cosx)]

 = [1*(1+cosx) / (1-cosx)*(1+cosx)] + [1*(1-cosx) / (1+cosx)*(1-cosx)]

 = [(1+cosx) / (1-cosx)(1+cosx)] + [(1-cosx) / (1+cosx)(1-cosx)]

Notice that the denominator in the first term, (1-cosx)(1+cosx), is equal to the denominator in the second term, (1+cosx)(1-cosx).

        (1-cosx)(1+cosx) = 1 + cosx - cosx - cos2x = 1 - cos2x  

Replace the denominators in both terms by this simplified term:

      [(1+cosx) / (1-cos2x)] + [(1-cosx) / (1-cos2x)]

Now that both fractions have a common denominator, we can add them:

       [(1+cosx)+(1-cosx)] / [(1-cos2x)]

    = (1+cosx+1-cosx) / (1-cos2x)

    =  2/(1-cos2x)

Recall the pythagorean identity:     sin2x + cos2x = 1

Subtracting cos2x from both sides of the equation, we arrive at:     sin2x = 1 - cos2x

Replace 1 - cos2x  by sin2x :

          2/(1-cos2x) = 2/sin2x

                           = 2*(1/sin2x)

Recall the following trig identity:      1/sinx = cscx ;  likewise,     1/sin2x = csc2x

Replace 1/sin2x with csc2x :

                                         2*(1/sin2x)  =  2*(csc2x)  =  2csc2x

Thus, we have proven that    1/(1-cosx) + 1/(1+cosx) = 2csc2x

Comments

Gina P. | Math and science tutorMath and science tutor
4.8 4.8 (34 lesson ratings) (34)
1

First we should try to simplify this by finding the LCM: 1/1-cosx + 1/1+cosX=

                 [1+cosx+1-cosx]/(1-cosx)(1+cosx). I got this by finding the LCM which is (1-cosx)*(1+cosx).

This simplifies to: 2/[1-(cosx)^2] which is the same as 2/[1-cos^2x]

Using pytagorean identities we know that sin^2x +cos ^2x = 1 so sin^2x = 1- cos^2x.

We can substitute sin^2x for (1-cos^2x) above and so we now have: 2/sin^2x

We know also that cosecx = 1/sinx

The above euation can be written as: 2[(1/sinx)(1/sinx)]. Substitute cosecx for 1/sinx and we get:

2(cosecx)(cosecx) this is the same as 2(cosecx)^2 which is equivalent to 2cosec^2x.

 

Hope that helps!

Gina

Robert J. | Certified High School AP Calculus and Physics TeacherCertified High School AP Calculus and Ph...
4.6 4.6 (13 lesson ratings) (13)
1

1/(1-cosx) + 1/(1+cosx)

= (1+cosx + 1-cosx)/(1-cos2x)

= 2/sin2x

= 2csc2x

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Attn: Use appropriate parenthesises to make your problem clear to read.