How do you solve #secxcscx - 2cscx = 0#?

1 Answer
Jul 28, 2015

Factorize the left hand side and equate the factors to zero.
Then, use the notion that : #secx=1/cosx" "# and #cscx=1/sinx#

Result : #color(blue)(x=+-pi/3+2pi"k , k"in ZZ )#

Explanation:

Factorizing takes you from
#secxcscx-2cscx=0#
to
#cscx(secx-2)=0#

Next, equate them to zero
#cscx=0=> 1/sinx=0#

However, there is no real value of x for which #1/sinx=0#

We move on to #secx-2=0#

#=>secx=2#

#=>cosx=1/2=cos(pi/3)#

#=>x=pi/3#

But #pi/3# is not the only real solution so we need a general solution for all the solutions.

Which is : #color(blue)(x=+-pi/3+2pi"k , k "in ZZ )#

Reasons for this formula:
We include #-pi/3# because #cos(-pi/3)=cos(pi/3)#

And we add #2pi# because #cosx# is of period #2pi#

The General solution for any #"cosine"# function is :

#x=+-alpha+2pi"k , k" in ZZ#

where #alpha# is the principal angle which just an acute angle

For example : #cosx=1=cos(pi/2)#

So #pi/2# is the principal angle!