Question

If sin A + Sin B = m and Cos A + Cos B = n .
Prove that :- Sin (A+ B) = 2mn / m²+n²

Answer 1

Answer:

sin A + Sin B = m and Cos A + Cos B = n

=2mn / m²+n²

putting value of m and n

=2×(sinA+sinB)/(sinA+cosB)^2+(cosA+cosB)^2

=2×(sinA+sinB)/(sin^2A+sin^2B+cos^2A+cos^2B)

=2×(sinA+sinB)/(sin^2A+cos^2A+sin^2B+cos^2A)

=2×(sinA+sinB)/(1+1)

=2×(sinA+sinB)/2

cancel 2 (2÷2)

=sinA+sinB

Hence proved.

OR

for easy method

take

A=B=45°

sin A+sin B= m

1÷√2+ 1÷√2=m

√2=m

cos A+cos B=n

1÷√2+ 1÷√2=n

√2=n

given

R.H.S

2mn÷[m²+n²]

=2×√2×√2÷[(√2)²+(√2)²]

=2×2÷2+2

=4÷4

= 1

L.H.S

=sin (A+B)

=sin (45°+45°)

=sin 90°

= 1

HENCE PROVED.

Explanation:

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