Expert: Ahmed Salami - 11/7/2004

Question
1.)  1/(cscx)(sec^2x) = ?

a.)1/(sinx)(cos^2x)
b.)sinx - sin^3x
c.)1/(sinx)(1+tan^2x)
d.)sinx - (1/1+tan^2x)
e.)1+tan^3x


2.)  solve the equation. find exact solutions when possible and approximate ones otherwise: cos 2x = cos x


I sincerely would like help on these questions because I'm struggling greatly on how to do them and any help would be much appreciated. Thanks a lot.


Answer
Hi Michael,
1)1/(cscx)(sec^2x) = [1/(cscx)].[1/(sec^2x)]
But 1/(cscx)= sinx and 1/(sec^2x)= cos^2x, therefore
1/(cscx)(sec^2x) = sinx.cos^2x
Also, cos^2x + sin^2x = 1 (this is an identity)
cos^2x = 1 - sin^2x
1/(cscx)(sec^2x) = sinx.cos^2x
                = sinx(1 - sin^2x)
                = sinx - sin^3x

2)cos2x = cosx
Using the idea of multiple angles,
cos2x = cos^2x - sin^2x
     = cos^2x - (1 - cos^2x)
     = 2cos^2x - 1
Therefore,
2cos^2x - 1 = cosx
which can be treated as a quadratic in cosx to give
2cos^2x - cosx - 1 = 0
with the solution
cosx = 1 or -1/2
this results in
x = 0,360 and 120,240
Note: The value gotten by adding 360 to any of the values for x are also solutions.
I hope you understand it.
Regards.