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# prove the identity tanx+cotx=secxcscx?

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- hsueh010Lv 79 years agoFavorite Answer
work from left side

sinx/cosx + cosx/sinx

common denominator

sin²x/(sinx cosx) + cos²x/(sinx cosx)

(sin²x + cos²x)/(sinx cosx)

Use identity that

(sin²x + cos²x) = 1

You have

1/(sinx cosx)

Split apart

1/sinx * 1/cosx

Or

1/cosx * 1/sinx

secx * cscx

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- PolyhymnioLv 79 years ago
tanx + cotx = sinx/cosx + cosx/sinx = (sin²x + cos²x)/(sinx cosx) =

1/(sinx cosx) = secx cscx

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- bored student 3Lv 59 years ago
tan x = sin x / cos x

cot x = 1/tan x = cos x /sin x

So, tan x + cot x = (sin x/cos x) + (cos x/sin x) = (sin^2 x + cos^2 x)/(sin x cos x)

sin^2 x + cos ^ x = 1

So, tan x + cot x = 1/(sin x cos x)

sec x = 1/cos x

csc x = 1/sin x

So, sec x csc x = (1/cos x)(1/sin x) = 1/(cos x sin x)

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