2cos-x-4-cos-x-4-

Question Number 9029 by rzagung last updated on 15/Nov/16

2 c o s (x + Π / 4) = c o s (x - Π / 4)

Answered by Rasheed Soomro last updated on 15/Nov/16

2 c o s (x + π / 4) = c o s (x - π / 4)

\cos (x + π / 4) + \cos (x + π / 4) - \cos (x - π / 4) = 0

\cos (x + π / 4) + (- 2 \sin \frac{(x + π / 4) + (x - π / 4)}{2} \sin \frac{(x + π / 4) - (x - π / 4)}{2}) = 0

\cos (x + π / 4) - 2 \sin (x) \sin (π / 4) = 0

\cos (x) \cos (π / 4) - \sin (x) \sin (π / 4) - 2 \sin (x) \sin (π / 4) = 0

\cos (x) \cos (π / 4) - 3 \sin (x) \sin (π / 4) = 0

\cos (x) (1 / \sqrt{2}) - 3 \sin (x) (1 / \sqrt{2}) = 0

\cos (x) - 3 \sin (x) = 0

\cos (x) = 3 \sin (x)

\frac{\sin (x)}{\cos (x)} = \frac{1}{3}

\tan (x) = \frac{1}{3}

x = \tan^{- 1} (\frac{1}{3})

x = n π + \tan^{- 1} (\frac{1}{3}) \forall n \in Z (in general)

Answered by Rasheed Soomro last updated on 15/Nov/16

2 c o s (x + π / 4) = c o s (x - π / 4)

\cos (x + π / 4) = \cos (x) \cos (π / 4) - \sin (x) \sin (π / 4)

= \frac{\cos (x)}{\sqrt{2}} - \frac{\sin (x)}{\sqrt{2}}

\cos (x - π / 4) = \cos (x) \cos (π / 4) + \sin (x) \sin (π / 4)

= \frac{\cos (x)}{\sqrt{2}} + \frac{\sin (x)}{\sqrt{2}}

2 (\frac{\cos (x)}{\sqrt{2}} - \frac{\sin (x)}{\sqrt{2}}) = \frac{\cos (x)}{\sqrt{2}} + \frac{\sin (x)}{\sqrt{2}}

2 (\cos (x) - \sin (x)) = \cos (x) + \sin (x)

\cos (x) = 3 \sin (x)

\frac{\sin (x)}{\cos (x)} = \frac{1}{3}

\tan (x) = \frac{1}{3}

x = \tan^{- 1} (\frac{1}{3}) + n π \forall n \in Z