Question Below, True or False?

Question

For all values for which the expressions are defined, sin x - cos x and

_____1_____ are equivalent expressions.
sin x + cos x

Answer 1

no.

take x = 45 degrees
then sin x - cos x = 0
while 1/(sinx + cosx) > 0

i think i understan a bit how you can get so far
(sin x - cos x ) * (sin x + cos x) = sin ^2x MINUS cos^2x
and 1 = sin^2 x PLUS cos ^ 2 x

Answer 2

To answer this question, simplt set the two expressions equal to each other. Play with the algebra, and if you eventually get something you know is false (like 2=1), then you know your premise (that they are equal) is false.

The contradiction is not hard to find. (sin+cos)(sin-cos)=sin^2-cos^2=1? Not true, except when sin= +/-1 and cos = 0 (x=pi/2 +k*pi, where k is an integer). Also, sin + cos is never equal to 0, so you don't even need to worry about things being defined, since you never divide by zero.

Answer 3

ok there are a few rules when doing this well not rules but stuff like sin^2x + cos^2x = 1 so really with the 1/sin x + cos x well I know they do equal its to hard to explain using this website

Answer 4

Let's see: sin(x) is an odd function, so sin(-x) = - sin(x)
and cos(x) is an even function, so cos(-x)= cos(x)

Note that sin(x) and cos(x) are defined on [0,2pi]

so, sin(x) - cos(x) = sin(x) - cos(-x) are equivalent.

Not sure the meaning of sin(x) +cos(x)...if this is your answer, it is incorrect.

Answer 5

true