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1962
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Problem 4
Problem 4
Find all real solutions of
cos
2
x
+
cos
2
2
x
+
cos
2
3
x
=
1
\cos^2 x+\cos^2 2x+\cos^2 3x=1
cos
2
x
+
cos
2
2
x
+
cos
2
3
x
=
1
.
Step 4 of 5: List the possible cosine values
Previous step
Next step
c
=
0
or
c
2
=
1
2
or
c
2
=
3
4
c=0\quad\text{or}\quad c^2=\frac12\quad\text{or}\quad c^2=\frac34
c
=
0
or
c
2
=
2
1
or
c
2
=
4
3
Detailed analysis
The factors give exactly
c
=
0
c=0
c
=
0
,
c
2
=
1
/
2
c^2=1/2
c
2
=
1/2
, or
c
2
=
3
/
4
c^2=3/4
c
2
=
3/4
.
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