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Simplify Expressions

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CPhill answered but he dropped the 2 on the -2cos4x.

Mar 30, 2018

#1
+1

Simplify: Sorry Adam, no LaTex !!
sin(x)^4 - sin(x)^2 cos(x)^2 - 2 cos(x)^4 =

Express sin(x)^2 in terms of cosine via the Pythagorean identity.
sin(x)^2 = 1 - cos(x)^2:
-2 cos(x)^4 - cos(x)^2 1 - cos(x)^2 + sin(x)^4 =

Expand -cos(x)^2 (1 - cos(x)^2).
-cos(x)^2 (1 - cos(x)^2) = cos(x)^4 - cos(x)^2:
-2 cos(x)^4 + cos(x)^4 - cos(x)^2 + sin(x)^4 =

Express sin(x)^4 in terms of cosine via the Pythagorean identity.
sin(x)^4 = (sin(x)^2)^2 = (1 - cos(x)^2)^2:
-2 cos(x)^4 - cos(x)^2 + cos(x)^4 + (1 - cos(x)^2)^2 =

Expand (1 - cos(x)^2)^2.
(1 - cos(x)^2)^2 = 1 - 2 cos(x)^2 + cos(x)^4:
-2 cos(x)^4 - cos(x)^2 + cos(x)^4 + 1 - 2 cos(x)^2 + cos(x)^4 =

Evaluate -2 cos(x)^4 - cos(x)^2 + cos(x)^4 + 1 - 2 cos(x)^2 + cos(x)^4.
-2 cos(x)^4 - cos(x)^2 + cos(x)^4 + 1 - 2 cos(x)^2 + cos(x)^4

= 1 - 3cos(x)^2

Mar 30, 2018
#4
+865
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Thanks Guest!

#2
+100571
+1

Sorry for the earlier error!!!

This becomes a little easier if we factor up front

sin^4x -sin^2xcos^2x -2cos^4x      factor as

(sin^2x  - 2cos^2x) (sin^2x + cos^2x)

(sin^2x - 2cos^2x)  (1)

sin^2x  - 2cos^2x

sin^2x  - 2 (1 - sin^2x)

3sin^2x -  2   or

3(1 - cos^2x) - 2

3 - 3cos^2x - 2

1 - 3cos^2x

Mar 30, 2018
#3
+865
+1

Thanks CPhill!

#5
+100571
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