Question

Prove geometrically that cos (x + y) = cos x cosy – sinx siny. 

Answer 1

Prove that cos(x + y) = cos x cos y .sin x sin y

Proot Consider a unit circle (radius = 1 unit) with centre is (0, 0). Consider 4 point and P₁, p₂, p₃ and p₄

The co-ordinate of and P₁, p₂, p₃ and p₄ are given by p₁ = (cosx,sinx) P₂ = [cos(x+y), sin(x+y)] 

p₃ = [cos(-y),sin (-y)] P₄ = [1,O] 

From the figure OP₁OP₃ is congruent to P₂P₄ 

∴ From distance formula p₁p₃ = p₂p₄ …(l) 

Take the distance (p₁P₃)² = [cosx – cos(-y)]² + [sin x – sin (-y)]² 

= (cosx – cosy)² + (sin x + sin y)² = cos²x + cos²y = cosxcosy + sin2x + sin2y + 2sincosy 

= 1 + 1 + 2(cosxcosy – sinxsiny) (P₁P₃)² = 2 – 2cos(x + y) 

Again (P₂P₄)² = [1(a-b)² -cos(x+y)]² + |q – sin (x + y)|² 

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

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

⇒ LHS = RHS ∴ [cos(x + y) = cosxcosy – sin x sin y] 

(ii) Show that cos2x = cos²x – sinx²x 

Take cos(x + y) = cosxcos y – sin xsin y 

Put y = X 

∴ cos(x + x) cosxcosx – sin xsin x 

cos2x = cos2x – sin2x