6. Prove that sin(A+B)=sinA.cosB+cosA.sinB Sin(A-B)= Cos(A-B)= Cos(A+B)=

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Q. Assertion :

\frac{sin (A + B) + sin (A - B)}{cos (A + B) + cos (A - B)} = tan A

Reason:

sin (A + B) + sin (A - B) = sin A

and

cos (A + B) + cos (A - B) = cos A

Q. Prove that

\frac{cos (A + B + C) + cos (- A + B + C) + cos (A - B + C) + cos (A + B - C)}{sin (A + B + C) + sin (- A + B + C) - sin (A - B + C) + sin (A + B - C)} = cot B

Q. Sin(A+B)+sin(A+B)/Cos(A+B)+cos(A-B)=TanA

Prove that:
(i) $\frac{s i n A + s i n 3 A}{c o s A - c o s 3 A} = c o t A$
(ii) $\frac{s i n 9 A - s i n 7 A}{c o s 7 A - c o s 9 A} = c o t 8 A$
(iii) $\frac{s i n A - s i n B}{c o s A - c o s B} = t a n \frac{A - B}{2}$
(iv) $\frac{s i n A + s i n B}{s i n A - s i n B} t a n (\frac{A + B}{2}) c o t \frac{A + B}{2}$
(v) $\frac{c o s A + c o s B}{c o s B - c o s A} = c o t \frac{A + B}{2} c o t \frac{A - B}{2}$

Q. Prove that:
(i)

\frac{\cos (A + B + C) + \cos (- A + B + C) + \cos (A - B + C) + \cos (A + B - C)}{\sin (A + B + C) + \sin (- A + B + C) + \sin (A - B + C) - \sin (A + B - C)} = \cot C

(ii) sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0

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