Hello, rtblue!
On the left side, we have: .
Multiply by
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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)