Question

cos( 3 pi/4+x )-cos( 3 pi/4-x)=-​

Answer 1

Question :

$\sf \: \cos \bigg( \dfrac{3 \pi}{4} + x\bigg) - \cos \bigg( \dfrac{3 \pi}{4} - x \bigg) =$

Solution :

First let us calculate the value of $\sf{\cos\big(\frac{3\pi}{4}-x\big)}$. By aplying the formulae Cos(A-B) = CosACosB + SinASinB . We get ;

$\\ : \implies \sf \: \cos \bigg( \frac{3 \pi}{4} - x \bigg) = \cos \bigg( \frac{3 \pi}{4} \bigg) . \cos(x) + \sin \bigg( \frac{3 \pi}{4} \bigg) . \sin(x)$

$\\ : \implies \sf \: \cos \bigg( \frac{3 \pi}{4} - x \bigg) = \cos \bigg( \frac{3 \times 180}{4} \bigg). \cos(x) + \sin \bigg( \frac{3 \times 180}{4} \bigg). \sin(x)$

$\\ : \implies \sf \: \cos \bigg( \frac{3 \pi}{4} - x \bigg) = \cos(135). \cos(x) + \sin(135) . \sin(x)$

$\\ : \implies \sf \: \cos \bigg( \frac{3 \pi}{4} - x \bigg) = - \frac{1}{ \sqrt{2} } . \cos(x) + \frac{1}{ \sqrt{2} } . \sin(x)$

$\\ : \implies \sf \: \cos \bigg( \frac{3 \pi}{4} - x \bigg) = \frac{ \sin(x) - \cos(x) }{ \sqrt{2} } \: .........(1)$

Now , Applying the formula i.e , Cos(A+B) = CosACosB - SinASinB. We get ;

$\\ : \implies \sf \: \cos \bigg( \frac{3 \pi}{4} \bigg) . \cos(x) - \sin \bigg( \frac{3 \pi}{4} \bigg). \sin(x) - \cos \bigg( \frac{3 \pi}{4} - x\bigg)$

$\\ : \implies \sf \: \bigg [ \cos \bigg( \frac{3 \times 180}{4} \bigg) . \cos(x) - \sin \bigg( \frac{3 \pi}{4} \bigg). \sin(x) \bigg] - \cos \bigg( \frac{3 \pi}{4} - x \bigg)$

$\\ : \implies \sf \bigg[ \cos(135). \cos(x) - \sin(135). \sin(x) \bigg] - \cos \bigg( \frac{3 \pi}{4} - x \bigg)$

$\\ : \implies \sf \bigg[ - \frac{1}{ \sqrt{2} } . \cos(x) - \frac{1}{ \sqrt{2} }. \sin(x) \bigg] - \cos \bigg( \frac{3 \pi}{4} - x \bigg)$

$\\ : \implies \sf \: \bigg[ - \frac{ \cos(x) }{ \sqrt{2} } - \frac{ \sin(x) }{ \sqrt{2} } \bigg] - \cos \bigg( \frac{3 \pi}{4} - x \bigg)$

$\\ : \implies \sf \bigg[ \frac{ - \cos(x) - \sin(x) }{ \sqrt{2} } \bigg] - \cos \bigg( \frac{3 \pi}{4} - x \bigg)$

From eq(1) ,

$\\ : \implies \sf \bigg[ \: \frac{ - \cos(x) - \sin(x) }{ \sqrt{2} } \bigg] - \bigg[ \frac{ \sin(x) - \cos(x) }{ \sqrt{2} } \bigg ]$

$\\ : \implies \sf \bigg[ \frac{ - \cos(x) - \sin(x) -[ \sin(x) - \cos(x) ] }{ \sqrt{2} } \bigg ]$

$\\ : \implies \sf \: \bigg[ \frac{ - \cos(x) - \sin(x) - \sin(x) + \cos(x) }{ \sqrt{2} } \bigg]$

$\\ : \implies \sf \bigg[ \frac{ - 2 \sin(x) }{ \sqrt{2} } \bigg]$

Rationalizing the denominator ,

$\\ : \implies \sf \: \frac{ - 2 \sqrt{2} \sin(x) }{ \sqrt{2} \times \sqrt{2} }$

$\\ : \implies \sf \: - \sqrt{2} \sin(x)$

Note :

$\\ \longmapsto \sf\cos(135) = - \frac{1}{ \sqrt{2} }$

$\\ \longmapsto \sf \: \sin(135) = \frac{1}{ \sqrt{2} }$

Answer 2

cos( 3 pi/4+x )-cos( 3 pi/4-x)= -√2 sinx.