Question

cos (x+y)= cos x cosy- sin x sin y​

Answer 1

Solution :

Consider a unit circle with centre O at the origin. Let A be the point (1, 0). Let P, Q and R be the points on the circle such that arc AP = x, arc PQ = y and arc AR = - y.

Then arc AQ = arc AP + arc PQ = x + y.

Therefore, the coordinates of the points P, Q and R are

(cos x, sin x), (cos (x + y), (sin (x + y)) and

(cos (-y), sin (-y)) respectively.

We have arc PQ = arc RA

arc PQ + arc AP = arc RA + arc AP

arc AQ = arc RP

length of chord AQ= length of chord RP

( In a circle, equal arcs cut off equal chords)

AQ = RP = AQ² = RP²

= (cos (r + y) - 1)² + (sin (x + y) - 0)² = (cos x - cos (-y))² + (sin x - sin (-y))²

= cos (x + y) +1 -2 cos (x + y) + sin² (x + y) = (cos x - cos y)² + (sin x + sin y)²

(•.•cos (-y) = cos y and sin (-y)= - sin y)

= (cos² (x + y) + sin² (x + y)+ 1 - 2 cos (x + y) = cos²x + cos²y - 2 cos x cos y +sin² x + sin² y + 2 sin x Sin y

= 1 + 1 - 2 cos (x + y) = (cos²x + sin² x) + (cos² y + sin2 y) - 2 cos x cos y + 2 Sin x sin y

= 2-2 cos (x + y) = 1 + 1 -2 cos x cos y +2 sin x sin y

= -2 cos (x + y) =-2 cos x Cos y + 2 sin x sin y

= cos (x + y) = cos x cos y - sin x siny.

Hence, cos (x + y) = cos x cos y - sin x sin y, for all x, y € R.

Answer 2

Given : -

• cos (x+y)= cos x cosy- sin x sin y

To Prove : -

• cos (x+y)= cos x cosy- sin x sin y

Solution : -

cosx cosy =1/2 ( cos( x + y) + cos( x - y)

sinx siny = 1/2 ( cos(x - y) - cos( x + y)

= cosx cosy - sinx siny = 1/2 ( cos( x + y) + cos(x - y) - cos( x - y) + cos(x + y)

= 1/2 ( 2 cos(x + y) )

= cos( x + y ) Proved

More information : -

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

.

B.Trigonometry Formulas involving Cofunction Identities (in Degrees)

sin(90°−x) = cos x

cos(90°−x) = sin x

tan(90°−x) = cot x

cot(90°−x) = tan x

sec(90°−x) = csc x

csc(90°−x) = sec x

C.Trigonometry Formulas involving Sum/Difference Identities:

sin(x+y) = sin(x)cos(y)+cos(x)sin(y)

cos(x+y) = cos(x)cos(y)–sin(x)sin(y)

tan(x+y) = (tan x + tan y)/ (1−tan x •tan y)

sin(x–y) = sin(x)cos(y)–cos(x)sin(y)

cos(x–y) = cos(x)cos(y) + sin(x)sin(y)

tan(x−y) = (tan x–tan y)/ (1+tan x • tan y)