Use an identity to solve the equation on the interval 021 ta

Use an identity to solve the equation on the interval [0,21). tan x sec x 1 Select the correct choice below and, if necessary, fill in the an

Solution

1)sec(x) = 1/cos(x)
tan(x) = sin(x)/cos(x)

sec(x) + tan(x) = 1
=> 1/cos(x) + sin(x)/cos(x) = 1
*Note here that by doing this step, cos(x) = 0 isn\'t allowed any more*
=> 1 + sin(x) = cos(x)
=> cos(x) + cos(x+pi/2) = 1

Which of course only happens when either:
- cos(x) = 0 and cos(x+pi/2) = 1, but cos(x) = 0 isn\'t allowed
- cos(x) = 1 and cos(x+pi/2) = 0

So cos(x) = 1 => x = 0 is the only solution (2pi is outside the given interval)

2) 20*sin^4 x

As sin^2 x = (1 - cos 2x)/2

sin^4 x = (1 + cos^2 2x - 2*cos 2x)/4

20*sin^4 x = 5*(1 + cos^2 2x - 2*cos 2x)

Now cos^2 2x=(1+cos 4x)/2

20*sin^4 x = 5*(1 - 2*cos 2x + 1/2 + (cos 4x)/2)

= (15/2) - 10*cos 2x + (5/2)*cos 4x