We have, L.H.S. = cos (

3π

4

+x) - cos (

3π

4

−x)
= 2 sin [

3π 4 +x+ 3π 4 −x

2

] sin [

( 3π 4 −x)−( 3π 4 +x)

2

]
Since cos A - cos B = 2 sin (

A+B

2

) sin (

B−A

2

)
= 2 sin (

3π

2

×

1

2

) sin (- x) = 2 sin (

3π

4

) sin (- x)
= - 2 sin (

4π−π

4

) sin x (Since sin (- x) = - sin x)
= - 2 sin (π−

π

4

) sin x = - 2 sin (

π

4

) sin x
[Since π−

π

4

lies in second quadrant in which sin x is +ve
= −

2

√2

sin x = - √2 sin x = R.H.S.