Question

If sin x + cos x = √2, then x =?​

Answer 1

★ Given :-

• sin x + cos x = √2

★ To find :-

• x = ?

★ Solution :-

given that

➟sin x + cos x = √2

squaring both sides

➟(sin x + cos x)^2 = (√2)^2

using identity (a+b)^2=a^2 + b^2 + 2ab

➟sin^2 x + cos^2 x + 2 sinx cosx = 2

using identity sin^2 A + cos^2 A = 1

➟1 + 2 sinx cos x = 2

➟ 2 sin x cos x = 2 - 1

➟ 2 sin x cos x = 1

dividing by cos^2 x both sides

➟(2 sinx cosx)/cos^2 x = 1 / cos^2 x

➟ 2 tan x = sec^2 x

using identity 1 + tan^2 A = sec^2 A

➟ 2 tan x = 1 + tan^2 x

➟ tan^2 x - 2 tan x + 1 = 0

putting tan x = m

➟ m^2 - 2 m + 1 = 0

finding the roots of quadratic eqn

➟ m^2 - m - m + 1 = 0

➟ m ( m - 1 ) -1 ( m - 1 ) = 0

➟ ( m - 1 ) ( m - 1 ) = 0

from here we get

➤ m = 1

since, tan x = m so, tan x = 1

➤ tan x = 1

hence, x = 45° ( Ans.)