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let y = (tan x)^{cot x} + (cot x)^{tan x}

⇒ y = a + b

where a= (tan x)^{cot x} ; b = (cot x)^{tan x}

a= (tan x)^{cot x}

Taking log both the sides:

⇒ log a= log (tan x)^{cot x}

⇒ log a= cot x log (tan x)

{log x^a = alog x}

Differentiating with respect to x:

b = (cot x)^{tan x}

Taking log both the sides:

⇒ log b= log (cot x)^{tan x}

⇒ log b= tan x log (cot x)

{log x^a = alog x}

Differentiating with respect to x: