can u solve this in less than 8 steps? double angle trig Identities

[struntz]

hello everyone!
i solved this trig identity but it took 8 steps! and i was thinking htere had to be an easier way and would like to find out...
here is what i did...
sec2x = sec^2x + sec^4x
----------------------
2 + sec^2x - sec%^4x

= sec^2x(1 + sec^2x)
-----------------------------
(2-sec^2x)(1+sec^2x)

= sec^2x
-----------
2-sec^2x

= 1/cos^2x
------------
2-1/cos^2x

= 1/cos^2x
---------------------------
2cos^2x - 1
-----------------
cos^2x

(1/cos2x)(cos^2x/2cos^2x -1)

= 1
-----------
2 cos^2x-1 //double angle, cos2x = 2cos^2x-1

= 1
--------
cos2x

= sec2x

if any of u can do it a differnet and faster way post it, thanks~!

[thinktank2]

(Sec[x])^2 + (Sec[x])^4
----------------------------
2 + (Sec[x])^2 - (Sec[x])^4


//--- Step 1 ---

= (Sec[x])^2 * [(Sec[x])^2 + 1]
-----------------------------------
[(Sec[x])^2 + 1] - [(Sec[x])^4 - 1]


//--- Step 2 ---
Dividing Both Numerator and Denominator by [(Sec[x])^2 + 1]

= [1 + (tan[x])^2] * 1
-------------------------
1 - [(Sec[x])^2 - 1]


//--- Step 3 ---

= [1 + (tan[x])^2]
---------------------
[1 - (tan[x])^2]


//--- Step 4 ---
Multiplying Both Numerator and Denominator by (Cos[x])^2

= (Cos[x])^2 + (Sin[x])^2
-----------------------------
(Cos[x])^2 - (Sin[x])^2


//--- Step 5 ---

= 1/Cos[2x]

//--- Step 6 ---

= Sec[2x]

[beachbum]

Step 1
Scroll down

Step 2
Copy

Step 3
Paste
Sec[2x]

done !

[NotLKH]

Step 1
Like BB Said!

Done!!!

Beat THAT!

[struntz]

Think tank, thanks for the responce, but i noticed u solved the proof as if it where

(Sec[x])^2 + (Sec[x])^4
----------------------------
2 + (Sec[x])^2 - (Sec[x])^4

but it really is
sec^2x + sec^4x
-----------------------
2 + sec^2x - sec^4x


or is that also correct becuasde u wrote it with the ( ) 's?

[Hyperspaced]

Well, it depends on what do you consider as a step. I didn't follow your proof but I think a proof with the steps detailed is this:


//Step 1// multiply by [sec(x)]^-4

1+[sec(x)]^-2
----------------------------------------
2[sec(x)]^-4 + [sec(x)]^-2 - 1


//Step 2// convert sec to cos

1 + [cos(x)]^2
--------------------------------------------
2 [[cos(x)]^2]^2 + [cos(x)]^2 - 1


//Step 3// factor denominator 2[w]^2 + w - 1 == (1+w)(2w-1)

1
---------------------
2[cos(x)]^2 - 1


//Step 4// known identity : 2[cos(x)]^2 - 1 = cos(2x)

1
--------
cos(2x)


Done !

Actually, copying & pasting is faster, but.....