SOLUTION: Consider the following equation: tan x + 1/tanx= secx/six a. verify that the equation is true for x = 2Π/3 b. Prove the equation is an identity

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Question 1193298: Consider the following equation:
tan x + 1/tanx= secx/six
a. verify that the equation is true for x = 2Π/3
b. Prove the equation is an identity
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
tan+%28x%29+%2B+1%2Ftan%28x%29=+sec%28x%29%2Fsin%28x%29
a. verify that the equation is true for x+=+2pi%2F3
tan%282pi%2F3%29%2B1%2Ftan%282pi%2F3%29=+sec%282pi%2F3%29%2Fsin%282pi%2F3%29

since
tan+%282pi%2F3%29=+-sqrt%283%29
sec%282pi%2F3%29=-2
sin%282pi%2F3%29=sqrt%283%29%2F2

substituting in equation above, we have

-sqrt%283%29+%2B+1%2F%28-sqrt%283%29%29=+-2%2F%28sqrt%283%29%2F2%29
%28-%28sqrt%283%29+%29%5E2%2B+1%29%2F%28-sqrt%283%29%29=+-4%2F%28sqrt%283%29%29
-%283%2B+1%29%2F%28-sqrt%283%29%29=+-4%2F%28sqrt%283%29%29
-4%2F%28-sqrt%283%29%29=+-4%2F%28sqrt%283%29%29 =>both sides same, the equation is true for x+=+2pi%2F3


b. Prove the equation is an identity

tan+%28x%29+%2B+1%2Ftan%28x%29=+sec%28x%29%2Fsin%28x%29...... use identity: sec%28x%29=1%2Fcos%28x%29

%28tan%5E2+%28x%29+%2B+1%29%2Ftan%28x%29=+%281%2Fcos%28x%29%29%2Fsin%28x%29



........sin%5E2%28x%29%2B+cos%5E2%28x%29=1

%281%2Fcos%28x%29%29%2F%28sin%28x%29%2F1%29=+1%2F%28cos%28x%29sin%28x%29%29

1%2F%28cos%28x%29sin%28x%29%29=+1%2F%28cos%28x%29sin%28x%29%29=>identity proven