How do you simplify the following?

Question

#(1-sin^2 x)/(sin x + 1)#

#(tan x)(1-sin^2 x)#

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Answer 1 · 2016-03-26T21:21:56

Example 1.

Use the difference of squares identity:

#a^2-b^2 = (a-b)(a+b)#

with #a = 1#, #b = sin x# as follows:

#(1-sin^2 x)/(sin x + 1) = ((1-sin x)color(red)(cancel(color(black)((1+sin x)))))/color(red)(cancel(color(black)((1+sin x)))) = 1 - sin x#

with exclusion #sin x != -1# (i.e. #x != (3pi)/2+2kpi#)

Example 2.

Use the following:

#sin^2 x + cos^2 x = 1# in the form #1 - sin^2 x = cos^2 x#

#tan x = (sin x)/(cos x)#

#sin 2x = 2 sin x cos x#

as follows:

#(tan x)(1 - sin^2 x) =(sin x)/(cos x)*cos^2 x = sin x cos x = 1/2 sin 2x#

with exclusion #cos x != 0#, i.e. #x != kpi#