Expert: Steve Holleran - 10/14/2007

Question
Hi, I'm pretty stuck on some questions I've been given.
The two main ones are,
a) Simplify tanA(1+cos2A)
b) Prove that 2+tan^2 A + cot^2 A =4/(sin^2 2A)

Thanks,
John

Answer
Hi John,

Okay, let's see what we can do here:

1.  Change cos 2a to 2cos^2 a - 1 using one of the double angle identities for cos 2a:

    tan a ( 1 + 2cos^2 a - 1)

=    tan a ( 2 cos^2 a)  = sin a / cos a * (2 cos^2 a)

cancel out cos a:    = sin a * 2 cos a

                    = 2 sin a cos a = sin 2a.

2.  This one involves a "trick":

Left side:

 Break the 2 into "1 + 1" and group one with the tan^2 a and the other with the cot^a:

  1 + tan^2 a + 1 + cot^2 a

=     sec^2 a    +   csc^2 a

=     1/ cos^2 a  + 1/ sin^2 a

common denominator:

        sin^2 a + cos^2 a   /  cos^2 a * sin^2 a

The top is an identity:

=              1   /  cos^2 a * sin^2 a

Multiply top and bottom by 4  (the trick)

=              4 / 4cos^2 a sin^2 a

write the bottom as a square:

              4 / (2cos a sin a)^2

the parenthesis is the double angle formula for sin 2a:

=            4 / sin^2 2a.

Hope this helps out.
Steve