Hey guys, I got stuck on this proof:

Prove that

Any help is appreciated!

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- April 11th 2011, 07:26 PMrtblueTrig Proof
Hey guys, I got stuck on this proof:

Prove that

Any help is appreciated! - April 11th 2011, 07:51 PMSoroban
Hello, rtblue!

Quote:

On the left side, we have: .

Multiply by

. .

. .

- April 11th 2011, 08:02 PMjohnny
sec x + tan x = 1/cos x + sin x / cos x = (1 + sin x)/cos x, cot x - csc x = cos x / sin x - 1/sin x = (cos x - 1)/sin x

(1 + sin x)/cos x =? (cos x - 1)/sin x

sin x + sin²x = cos²x - cos x

1 = cos x - sin x ∴ 1 + sin x = cos x, sin x = cos x - 1

∴ sec x + tan x = cot x - csc x = 1

cot x - csc x + 2 csc x = sec x + tan x + 2 csc x

2 csc x =? -2 sec x

csc x =? -sec x

ASTC for four quadrants, where sine for second quadrant and cosine for fourth quadrant

∴ csc x = -sec x

∴ 2 csc x = -2 sec x

∴ cot x - csc x + 2 csc x = sec x + tan x + 2 csc x = sec x + tan x - 2 sec x = tan x - sec x

∴ (sec x + tan x)/(csc x + cot x) = (cot x - csc x)/(tan x - sec x)