# If X = E Cos 2 T and Y = E Sin 2 T , Prove that D Y D X = − Y Log X X Log Y . - Mathematics

Sum

If "x" = "e"^(cos2"t")  "and"  "y" = "e"^(sin2"t"), prove that (d"y")/(d"x") = - ("y"log"x")/("x"log"y").

#### Solution

Here "x" = "e"^(cos2"t") and  "y"="e"^(sin2"t")

⇒ log_"e" "x" = log_"e" ("e"^(cos2"t")) and log_"e" "y" = log_"e"("e"^(sin2"t"))

⇒ log_"e" "x" = cos 2"t" log_"e" ("e") and log_"e" "y"=sin 2"t" log_"e" ("e")    ...["As log"_"e" ("e") = 1]

∴ log_"e" "x" = cos 2"t" and log_"e" "y" = sin 2"t"

Squaring and then adding these two equations,

(log_"e" "x")^2 + (log_"e" "y")^2 = cos^2 2"t" + sin^2 2"t"

⇒ (log_"e" "x")^2 + (log_"e" "y")^2 = 1

⇒ 2(log_"e" "x") xx (1)/("x") + 2(log_"e" "y") xx (1)/("y") xx (d"y")/(d"x") = 0

∴ (d)/(d"x") [(log_"e" "x")^2 + (log_"e" "y")^2] = (d)/(d"x")  ...(1)

⇒ (log_"e" "y")/("y") xx (d"y")/(d"x") = - (log_"e" "x")/("x")

∴ (d"y")/(d"x") = - ("y"log"x")/("x"log"y")

Concept: Exponential and Logarithmic Functions
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