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Johanna Dickerson
Mathematics
23 May, 02:53
Verify Sin^4x+2cos^2x-cos^4=1
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Kendall Terry
23 May, 02:59
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Proven. We get a true statement of 1 = 1 by transforming the expression on the left side to make it look like the right side. See below.
Step-by-step explanation:
This is missing some notation: Sin^4x+2cos^2x-cos^4=1
We want to prove : (Sin x) ^ 4 + 2 (cos x) ^2 - (cos x) ^4 = 1
Replace the (cos x) ^4 with ((cos x) ^2) ^2 same with the (sin x) ^4 with
((sin x) ^2) ^2
(Sin x) ^ 4 + 2 (cos x) ^2 - (cos x) ^4 = 1
(((sin x) ^2) ^2 - ((cos x) ^2) ^2 + 2 (cos x) ^2 + 1 - 1 = 1
Factor the trinomial - ((cos x) ^2) ^2 + 2 (cos x) ^2 + 1.
considering ((cos x) ^2) is the variable
(((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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