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Johan Hendrix
Mathematics
25 February, 07:10
Sin^4x+2cos^2x-cos^4=1
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Alice Moses
25 February, 07:14
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Proven. We are done proving this identity because we get a true statement, which is 1 = 1.
Step-by-step explanation:
Sin^4x+2cos^2x-cos^4=1
...
We want to prove : (Sin x) ^ 4 + 2 (cos x) ^2 - (cos x) ^4 = 1
(Sin x) ^ 4 + 2 (cos x) ^2 - (cos x) ^4 = 1
Factor this trinomial. considering ((cos x) ^2) is the variable
so
(((sin x) ^2) ^2 - ((cos x) ^2) ^2 + 2 (cos x) ^2 + 1 - 1 = 1
(((sin x) ^2) ^2 - ((cos x) ^2) ^2 + 2 (cos x) ^2 - 1 + 1 = 1
(((sin x) ^2) ^2 + [ - ((cos x) ^2) ^2 + 2 (cos x) ^2 - 1 ] + 1 = 1
(((sin x) ^2) ^2 - [ ((cos x) ^2) - 1 ]^2 + 1 = 1
But also notice that (sin x) ^2 = 1 - (cos x) ^2 from the trig identity:
(sin x) ^2 + (cos x) ^2 = 1
((1 - (cos x) ^2) ^2 - [ ((cos x) ^2) - 1 ]^2 + 1 = 1
here we see that (1 - (cos x) ^2) ^2 = [ ((cos x) ^2) - 1 ]^2
so we get (0 + 1) = 1
1 = 1 true.
Proven. We are done proving this identity because we get a true statement.
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