Question

if cosA+sinA=√2cosA , then prove that cosA-sinA=√2sinA​

Answer 1

$\large\bf\underline\red{Question \: :- }$

★ If cosA+sinA=√2cosA , then prove that cosA-sinA=√2sinA

$\large\bf\underline\red{Answer \: :- }$

Given,

$\sf{cosA + sinA = \sqrt{2} cosA }$

Replacing

$\sf{sinA = \sqrt{2} cosA - cosA}$

$\sf{sinA = ( \sqrt{2} - 1 ) \: cosA}$

$\sf{ \frac{\sin A}{cosA} = ( \sqrt{2} -1)}$

$\sf{tanA = \sqrt{2} - 1 }$

$\sf{ \frac{1}{tanA} = \frac{1}{ \sqrt{2} - 1 } }$

Rationalize denominator

$\sf{cotA = \frac{1}{ \sqrt{2} - 1} \times \frac{ \sqrt{2} + 1}{ \sqrt{2} + 1 } }$

$\sf{cotA = \sqrt{2} + 1 }$

We know that

$\sf{cotA = \frac{cosA }{ sinA} }$

Putting

$\sf{ \frac{cosA }{ sinA} = \sqrt{2} + 1 }$

$\sf{cosA = sinA ( \sqrt{2} + 1) }$

$\sf{cosA = \sqrt{2} sinA + sinA }$

$\sf\pink{cosA - sinA = \sqrt{2} sinA}$

Hence Proved

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