how can i prove this?!?

Question

sec²(x)+csc²(x)=(tan(x)+cot(x))²

yea that's weird, yahoo cutoff the question
its supposed to be:
sec²(x) + csc²(x) = (tan(x)+cot(x))²

Answer 1

simple

1/sec(x)= cos(x)
1/csc(x)= sin(x)
sin^2(x) + cos^2(x)=1
2sin(x)cos(x)=sin(2x)

so

(1/cos^2(x) +1/sin^2(x))=
4/(sin^2(2x)) is not close to (tan(x)+cot(x)...
so what is missing?

Using additional info we have
tan = sin/cos
cot = cos/sin
(tan(x)+cot(x))^2=(sin(x)/cos(x) + cos(x)/sin(x))^2=
=[(cos^2(x)+sin^2(x)]/sin(x)cos(x))^2=
=(1/sin(x)cos(x))^2=
=(2/sin(2x))^2=
=4/sin^2(2x)

Mistake corrected.
Proof is compleat.
Have fun

Answer 2

Start by substituting in sine and cosine for the functions and simplifying. For example the left side of what you wrote becomes ...

sec^2 + csc^2
1/cos^2 + 1/sin^2
(sin^2 + cos^2)/ (sin^2cos^2)
1 / (sin^2cos^2)

Since your problem is not finished, this is the best advice I can give.

Answer 3

Ugh, not one right answer. We'll start with the right hand side:

(tan x + cot x)²
Expand the terms:
(sin x/cos x + cos x/sin x)²
Find a common denominator:
(sin² x/(sin x cos x) + cos² x/(sin x cos x))²
Add:
((sin² x + cos² x)/(sin x cos x))²
sin² x + cos² x = 1, so:
(1/(sin x cos x))²
Multiply:
1/(sin² x cos² x)
sin² x + cos² x = 1, so:
(sin² x + cos² x)/(sin² x cos² x)
Split the fraction:
sin² x/(sin² x cos² x) + cos² x/(sin² x cos² x)
Cancel:
1/cos² x + 1/sin² x
And this is:
sec² x + csc² x

Answer 4

What do you mean by (tan(x)+cot(x)... ?

I could find only one similar transformation: using sec(x) = [1/cos(x) and csc(x)= 1/sin(x)

[1/cos(x)]² + [1/sin(x)]²
= [cos(x) ² + sin(x) ² ] / [cos(x)²sin(x)²]
=cos(x) ² / [cos(x)²sin(x)²] + sin(x) ² / [cos(x)²sin(x)²]
= [cot(x)/cos(x)]² + [tan(x)/sin(x)]²

Answer 5

Your problem is missing something, I think. You have to put spaces or Yahoo cuts things off.

Answer 6

If you believe in it there is no need to prove it.