How do you prove #sin(x) x tan(x) + cos(x) = sec(x)#?
1 Answer
Feb 8, 2016
I strongly assume that you would like to prove the identity
#sin(x) tan(x) + cos(x) = sec(x)#
without the
(Maybe it was a "times" that was confused with "
===============================================
We will need the following identities:
[1]
#" " tan(x) = sin(x)/cos(x)# [2]
#" " sec(x) = 1/cos(x)# [3]
#" " sin^2(x) + cos^2(x) = 1#
Let's start at the left side and try to get to the right side:
#sin(x) tan(x) + cos(x) stackrel("[1] ")(=) sin(x) * sin(x)/cos(x) + cos(x)#
# = sin^2(x)/cos(x) + cos(x) * color(blue)(cos(x)/cos(x))#
# = sin^2(x)/cos(x) + cos^2(x) / cos(x)#
# = (sin^2(x) + cos^2(x)) / cos(x)#
# stackrel("[3] ")(=) 1 / cos(x)#
# stackrel("[2] ")(=) sec(x)#
q.e.d.
