Express as a sum or a difference 2 sin 7 theta sin 5 theta E

Express as a sum or a difference. 2 sin 7 theta sin 5 theta Express as product. sin 8t + sin 2t Find the exact value of the expression whenever it is defined. cot (sin^-1 2/3)

Solution

2sin 7theta sin 5theta

cos A- cos B= 2sin (A+B)/2 sin(A-B)/2

On comparing the two expressions we get

(A+B)/2=7 theta   and (A-B)/2=5theta

On solving these two equations we get A= 12 theta and B=12 theta

Therefore 2sin 7theta sin 5theta=cos 12 theta -cos 2theta

7. sin8t + sin2t

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

therefore sin8t + sin2t= 2sin(8t+2t)/2 cos (8t-2t)/2= 2sin5tcos3t

8. cot(sin^-12/3)

let sin^-12/3=t

sin t=2/3

sin theta=opposite/hypotenuse

adjacent=sqrt(hypotenuse²-opposite²)= sqrt(9-4)=sqrt5

cot t= adjacent/opposite= sqrt5/2

therefore cot(sin^-12/3)=sqrt5/2