# 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 "

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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.