How do you verify #tan(x)sin(x) + cos(x)= sec(x) #?

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Answer 1 · 2016-04-14T01:22:43

Recall the following quotient, Pythagorean, and reciprocal identities:

#1. color(red)(tanx=sinx/cosx)#

#2. color(darkorange)(sin^2x+cos^2x=1)#

#3. color(blue)(secx=1/cosx)#

#1#. To verify the given identity, start by working on the left side. Rewrite #tanx# in terms of #sinx# and #cosx#.

Left side:

#color(red)tanxsinx+cosx#

#=color(red)(sinx/cosx)*sinx+cosx#

#2#. Simplify.

#=sin^2x/cosx+cosx#

#=color(darkorange)(sin^2x+cos^2x)/cosx#

#3#. Simplify the numerator.

#=color(darkorange)1/cosx#

#=color(green)(|bar(ul(color(white)(a/a)secxcolor(white)(a/a)|)))#