Solve the following equation for
0 less than and/or equal to "x" less than and/or equal to 360
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cos^2x - 1 = sin^2x
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Attempt:
cos^2x - 1 - sin^2x = 0
cos^2x - 1 - (1 - cos^2x) = 0
cos^2x - 1 - 1 + cos^2x = 0
2cos^2x - 2 = 0
(2cos^2x/2)= (-2/2)
cos^2x = -1
cosx = square root of -1
And I can't do anything with this now...what am I doing wrong?
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Textbook answers:
0, 180, 360
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On the fourth/fifth line, you erred. When you add 2 to both sides, the right side is positive, not negative.
cos^2 x=1
cos x= +- 1
giving 1, 180,360
To solve the equation cos^2x - 1 = sin^2x, you need to use the trigonometric identity that relates the sine and cosine functions.
The identity you can use is: sin^2x + cos^2x = 1
Start by substituting sin^2x with 1 - cos^2x, as per the identity.
So the equation becomes: cos^2x - 1 = 1 - cos^2x
Rearrange the equation by moving all the terms on one side: cos^2x - cos^2x - 1 + 1 = 0
Combine like terms: 0 = 0
This equation is true for any value of x within the given range 0 <= x <= 360. Therefore, the solution is x = 0, x = 180, and x = 360.