My First problem on Trigonometry.......

$\frac{tanx + cotx}{2} \geq \sqrt{tanx\times cotx}$

its least value is given thus tanx = cotx

$x = \frac{\pi}{4}$

$tan^{7}x + cot^{7}x = 2$

Apart from Sandeep's AM-GM, Simply observe that 1/tan = cot, Therefore multiplying by by tan x, you get a quadratic, solve and get tan x = 1 And the rest is trivial...

Re write equation as tan(t) + 1/(tan(t)) = 2, or tan^2(t) + 1 = 2tan(t), sec^2(t) = 2 tan(t), 1/[cos^2(t)] = 2sin(t)/cos(t), sin(2t) = 1,, t = pi/4, tan(pi/4) = 1, cot(pi/4) = 1, tan^7(t) + cot^7(t) = 1 + 1 = 2. Ed Gray

Tanx +1/tanx=2 is satisfied only by tanx since for all real non zero x. x+1/x>=2.thus Tanx = 1 and cot x=1.thus tan^7x+cot^7x= 1

no matter what exponents it is to be raised theta will always be 45 leads to 1 summing up = 2