Sine of the Difference and Sum of Two Angles

Sine of the Difference and Sum of Two Angles

1. sin (A-B) = sinA cosB + cosA sinB

Proof: L.H.S: sin (A-B)

= sinA cosB + cosA sinB

L.H.S sin (A – B) = cos (90 – (A – B))

(since cos (90 – θ) = sin θ)

= cos (90 – (A – B))

= cos [(90 – A) + B]

= cos (90° – A) cos B – sin (90° – A) sin B

(since cos (90 – θ) = sin θ, sin (90 – θ) = cos θ)

= sin A cos B – cos A sin B

sin(A-B) = sinA cosB + cosA sinB

Hence proved

2. sin(A+B) = sinA cosB + cosA sinB

Proof: L.H.S: Sin (A + B)

 = sin (A – (-B))

= sinA cos(-B) – cosA sin(-B)      

(since cos(-θ) = cosθ

and sin(-θ) = -sinθ)

= sinA cosB + cosA sinB

sin(A+B) = sinA cosB + cosA sinB

Hence proved

3. sin (A + B) sin (A – B) = sin²A – sin²B

Proof: L.H.S: Sin (A + B) sin (A – B)

= (sinA cosB + cosA sinB) (sinA cosB – cosA sinB)

= sinA cosB x sinA cosB – sinA cosB x cosA sinB + cosA sinB x sinA cosB – cosA sinB x cosA sinB

= sinA cosB x sinA cosB – cosA sinB x cosA sinB

= sin²A cos²B – cos²A sin²B

= sin²A (1-sin²B) – (1-sin²A) sin²B

= sin²A – sin²A x sin²B – sin²B + sin²A x sin²B

= sin²A – sin²B

Sin (A + B) sin(A – B) = sin²A – sin²B

Hence proved.