Question

prove that
1. sin(A - B) = sinAcos B - cos AsinB
2. cos(A + B) = cos AcosB – sin Asin B
3. cos(A - B) = cos AcosB + sinAsinB
4. 2tan B= tan2B /1- tan2B​

Answer 1

Answer:

sinB

sinA

=p;

cosB

cosA

=q

sinA=psinB−−−(1);cosA=qcosB−−−(2)

tanA=

q

p

tanB−−−(3)(dividing(1)by(2))sinAcosA=pqsinBcosB(Multiply(1)&(2))

cos

2

Acos

2

B

sinAcosA

=

cos

2

Acos

2

B

pqsinBcosB

=sec

2

BtanA=pqsec

2

AtanB

=(1+tan

2

B)tanA=pq(1+tan

2

A)tanB

=[1+(

q

p

tanA)

2

]tanA=pq(1+tan

2

A).

q

p

tanA(by(3))

=1+

q

2

p

2

tan

2

A=q

2

+q

2

tan

2

A

=tan

2

A(

p

2

q

2

−q

2

)=q

2

−1

=tan

2

A

q

2

−p

2

q

2

(q

2

−1)p

2

tanA=

q

2

(p

2

−1)

p

2

(1−q

2

)

=

q

p

p

2

−1

1−q

2

tanB=

p

2

−1

1−q

2