Prove x -2 n -2 or x -2 n - where n - I-

Cos x +sin x =1 ⇒1/√(2)cos x +1/√(2)sin x =1/√(2) ⇒ cos x cos π/4+sin x sin π/4=1/√(2) ⇒ cos ( x π/4 )=cos π/4 ⇒ x π/4=2 n π±π/4 ⇒ x π/4=2 n π +π/4 or x ...

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Standard XII

Mathematics

Range of Trigonometric Expressions

Prove x =2 n ...

Question

Prove $x = (2 n π + \frac{π}{2})$ or $x = 2 n π$ , where $n \in I$ .

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Solution

$c o s x + s i n x = 1 \Rightarrow \frac{1}{\sqrt{2}} c o s x + \frac{1}{\sqrt{2}} s i n x = \frac{1}{\sqrt{2}}$

$\Rightarrow c o s x c o s \frac{π}{4} + s i n x s i n \frac{π}{4} = \frac{1}{\sqrt{2}}$

$\Rightarrow c o s (x - \frac{π}{4}) = c o s \frac{π}{4}$

$\Rightarrow x - \frac{π}{4} = 2 n π \pm \frac{π}{4}$

$\Rightarrow x - \frac{π}{4} = 2 n π + \frac{π}{4}$ or $x - \frac{π}{4} = 2 n π - \frac{π}{4}$

$\Rightarrow x = 2 n π + \frac{π}{2}$ or $x = 2 n π$ , where $n \in I$ .

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Prove x=(2nπ+π2) or x=2nπ, where n∈I.

cos x+sin x=1⇒1√2cos x+1√2sin x=1√2 ⇒cosxcosπ4+sinxsinπ4=1√2 ⇒cos(x−π4)=cosπ4 ⇒x−π4=2nπ±π4 ⇒x−π4=2nπ+π4 or x−π4=2nπ−π4 ⇒x=2nπ+π2 or x=2nπ, where n∈I.

Prove $x = (2 n π + \frac{π}{2})$ or $x = 2 n π$ , where $n \in I$ .