(Solved) You are given the trig equation: csc^2 x + sec^2 x = (csc^2 x) (sec^2 x).

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10 years ago

You are given the trig equation: csc^2 x + sec^2 x = (csc^2 x) (sec^2 x).

You are given the trig equation: csc^2 x + sec^2 x = (csc^2 x) (sec^2 x).

a. State all restrictions on x

b. Rewrite the two sides of this identity using only sine and cosine functions.

c. Complete the proof of this identity using the results of the previous bullet.

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skip5284

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10 years ago

sometimes, we can starting from the right side.
we knowed that
csc = 1/sin and sec = 1/cos

so,
csc^2(x) sec^2(x)
= 1/sin^2(x) * 1/cos^2(x)
= 1/(sin^2x cos^2 x)

remmber this identity :
1 = sin^2x + cos^2x

therefore,
1/(sin^2x cos^2 x)
= (sin^2(x) + cos^2(x))/(sin^2x cos^2 x)
splite into 2 parts, get
= sin^2x/(sin^2x cos^2 x) + cos^2x/(sin^2x cos^2 x)
= 1/cos^2x + 1/sin^2(x)
= sec^2(x) + csc^2(x)

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chemb

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10 years ago

csc^2 x + sec^2 x = csc^2 x * sec^2 x

We will start from the left side and prove the right side.

We know that:

csc(x) = 1/sin(x) ==> csc^2 x = 1/sin^2 x

sec(x) = 1/cos(x) ==> sec^2 x = 1/cos^2 x

We will substitute:

==> (1/sin^2 x) + (1/cos^2 x)

Now we will rewrite using the common denominator (sin^2 x*cos^2 x)

==> (cos^2 x + sin^2 x) / (sin^2 x* cos^2 x)

Now we know that sin^2 x + cos^2 x = 1

==> 1 / (sin^2 x *cos^2 x)

==> 1/sin^2 x * 1/cos^2 x

==> csc^2 x * sec^2 x ...................q.e.d

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mikael

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You are given the trig equation: csc^2 x + sec^2 x = (csc^2 x) (sec^2 x).

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