Question

sec²x - 2tanx = 0
find value of x.​

Answer 1

AnswEr :

x could be π/4 or 3π/4

Given,

$\sf \: {sec}^{2} x - 2tan \: x = 0$

But sec²x = 1 + tan²x

$\longrightarrow \sf {tan}^{2} x - 2tan \: x + 1 = 0 \\ \\ \longrightarrow \sf \: (tan \: x) {}^{2} - 2(tan \: x)(1) + (1) {}^{2} = 0 \\ \\ \longrightarrow \sf \:( tan \: x - 1) {}^{2} = 0 \\ \\ \longrightarrow \: \sf \: tan \: x = 1 \\ \\ \longrightarrow \sf \: tan \: x = tan \: \dfrac{\pi}{4} \\ \\ \longrightarrow \: \boxed{ \boxed{ \sf \: x = \dfrac{\pi}{4}}}$

Also,

$\sf \: tan \: x = tan(\pi - \dfrac{\pi}{4} ) \\ \\ \longrightarrow \sf \: tan \: x = tan( \dfrac{3\pi}{ 4} ) \\ \\ \longrightarrow \: \boxed{ \boxed{ \sf \: x = \dfrac{3\pi}{4}}}$

General Solution

$\large{ \sf \: x = n \pi + \dfrac{\pi}{4}}$