Expert: Steve Holleran - 2/26/2007

Question
how would you solve these two identities step by step?
1. (sec X/ sin X)-(sin X/ cos X)=cot X

2. (sin X + cos X)^2= 1+ sin2x

Answer
Hello Michael,

1.  Working on the left side of this one, first let's get a common denominator:  sin x * cos x.  Then, combining the two fractions into one large fraction over the common denominator, we get:

     [cos x * sec x - sin x * sin x] / [sin x * cos x]

  =  [cos x * 1/cos x - sin^2 x] / [ sin x * cos x]

  =  [1 - sin^2 x ] / [sin x * cos x]

  =    [ cos^2 x ] / [sin x * cos x]   (using  an identity
                                        for 1-sin^2 x)

  =  cos x / sin x     (cancelling out a factor of cos x)

  = cot x

2.  On this one, you want to square out the left hand side:

(sin x + cos x)^2 = sin^2 x + 2 sin x * cos x + cos^2 x

Now group the sin^2 x + cos ^2 x together and by identity, this equals 1.  so now you have:

   1 + 2 sin x * cos x

and the term 2 sinx * cos x is the identity for sin (2x),

so you have 1 + sin (2x) and you're done.

I hope this helps you out.