Solution for tan x is equal to cos x?

Answer 1

Tan(x) = Sin(x) / Cos(x)

Hence

Sin(x) / Cos(x) = Cos(x)

Sin(x) = Cos^(2)[x]

Sin(x) = 1 - Sin^(2)[x]

Sin^(2)[x] + Sin(x) - 1 = 0

It is now in Quadratic form to solve for Sin(x)

Sin(x) = { -1 +/-sqrt[1^(2) - 4(1)(-1)]} / 2(1)

Sin(x) = { -1 +/-sqrt[5[} / 2

Sin(x) = {-1 +/-2.236067978... ] / 2

Sin(x) = -3.236067978...] / 2

Sin(x) = -1.61803.... ( This is unresolved as Sine values can only range from '1' to '-1')

&

Sin(x) = 1.236067978... / 2

Sin(x) = 0.618033989...

x = Sin^(-1) [ 0.618033989...]

x = 38.17270765.... degrees.

Answer 2

if tan x = cos x then

sin x / cos x = cos x

=> sin x = cos x cos x

=> sin x = cos2 x

=> sin x = 1 - sin2x

=> sin2x + sin x - 1 = 0

Using the quadratic formula

=> 1. sin x = 0.61803398874989484820458683436564

=> x = sin-1 (0.61803398874989484820458683436564)

or

=> 2. sin x = -1.6180339887498948482045868343656

=> x = sin-1 (-1.6180339887498948482045868343656)