Answer:
cos x cos (-x) -sin x sin (-x) = 1 ⇒ proved down
Explanation:
* Lets revise the angles in the four quadrants
- If angle x is in the first quadrant, then the equivalent angles to it are
# 180 - x ⇒ second quadrant (sin (180 - x) = sin x , cos (180 - x) = -cos x
tan (180 - x) = -tan x)
# 180 + x ⇒ third quadrant (sin (180 - x) = -sin x , cos (180 - x) = -cos x
tan (180 - x) = tan x)
# 360 - x ⇒ fourth quadrant (sin (180 - x) = -sin x , cos (180 - x) = cos x
tan (180 - x) = -tan x)
# -x ⇒fourth quadrant (sin (- x) = -sin x , cos (- x) = cos x
tan (- x) = -tan x)
* Lets solve the problem
∵ L. H .S is ⇒ cos x cos (-x) - sin (x) sin (-x)
- From the rules above cos x = cos(-x)
∴ cos x cos (-x) = cos x cos x
∴ cos x cos (-x) = cos² x
- From the rule above sin (-x) = - sin x
∴ sin x sin (-x) = sin x [- sin x]
∴ sin x sin (-x) = - sin² x
∴ cos x cos (-x) - sin (x) sin (-x) = cos² x - (- sin² x)
∴ cos x cos (-x) - sin (x) sin (-x) = cos² x + sin² x
∵ cos² x + sin² x = 1
∴ R.H.S = 1
∴ L.H.S = R.H.S
∴ cos x cos (-x) -sin x sin (-x) = 1