Question

If $cos\theta +sin\theta =\sqrt{cos\theta }$, then what is the value of $cos\theta -sin\theta$? [3 MARKS]

Answer 1

Formula: 1 Mark
Application: 2 Marks

$cos\theta +sin\theta =\sqrt{2}cos\theta$

$(cos\theta +sin\theta {)}^2=(\sqrt{2}cos\theta {)}^2$

$co{s}^2\theta +si{n}^2\theta +2sin\theta cos\theta =2co{s}^2\theta$

$1+2sin\theta cos\theta =2co{s}^2\theta$

$2sin\theta cos\theta =2co{s}^2\theta -1$ ..........(1)

$(cos\theta -sin\theta {)}^2=co{s}^2\theta +si{n}^2\theta -2sin\theta cos\theta$

$=1-2sin\theta cos\theta$

$=1-2co{s}^2\theta +1$

$=2-2co{s}^2\theta$

$=2(1-co{s}^2\theta )$

$=2si{n}^2\theta$

$(cos\theta -sin\theta {)}^2=2si{n}^2\theta$

$cos\theta -sin\theta =\sqrt{2}sin\theta$