Question

$(\csc( \alpha ) - \csc( \beta ) ) ^{2} + ( \sin( \alpha ) - \sin( \beta ) )^{2} = 4 \sin^{2} \frac{ \alpha - \beta }{2}$
​

Answer 1

Answer:

There is a correction in your question.

Step-by-step explanation:

alpha = A & beta = B

Here,

=> (cosA - cosB)² + (sinA - sinB)²

=> cos²A + cos²B - 2cosAcosB + sin²A + sin²B - 2sinAsinB

=> (cos²A + sin²A) + (cos²B + sin²B) - 2cosAcosB - 2sinAsinB

=> 1 + 1 - 2cosAcosB - 2sinAsinB

=> 2 - 2(cosAcosB + sinAsinB)

{cosAcosB+sinAsinB=cos(A-B)}

=> 2 - 2cos(A - B)

=> 2(1 - cos(A - B))

{1 - cosx = 2sin²(x/2)}

=> 2(2sin²(A-B)/2}

=> 4sin²(A-B)/2 proved