Question

Tan (A+30°) +Cot (A-30°) =1/Sin2A-Sin60°

Answer 1

tan(a+30)+cot(a-30) = 1/sin 2a - sin60

take LHS

tan(a+30) +cot(a+30)

>> = sin(a+30)/cos(a+30) + cos(a-30)/sin(a-30)

taking LCM

>> = sin(a+30)sin(a-30) + cos(a+30)cos(a-30)/cos(a+30)sin(a-30)

using cos A cos B+sin A sin B = cos(A-B) (in numerator)

>> = cos(a+30 - (a-30))/cos(a+30)sin(a-30)

>> = cos(60)/cos(a+30)sin(a-30)

putting cos60=1/2

>> = 1/2cos(a+30)sin(a-30)

but 2cosAsinB = sin 2 A - sin 2 B

>> = 1/sin 2 a - sin 2(30)

>> = 1/sin 2a - sin60

Answer 2

Step-by-step explanation:

tan(a+30)+cot(a-30) = 1/sin 2a - sin60

take LHS

tan(a+30) +cot(a+30)

>> = sin(a+30)/cos(a+30) + cos(a-30)/sin(a-30)

taking LCM

>> = sin(a+30)sin(a-30) + cos(a+30)cos(a-30)/cos(a+30)sin(a-30)

using cos A cos B+sin A sin B = cos(A-B) (in numerator)

>> = cos(a+30 - (a-30))/cos(a+30)sin(a-30)

>> = cos(60)/cos(a+30)sin(a-30)

putting cos60=1/2

>> = 1/2cos(a+30)sin(a-30)

but 2cosAsinB = sin 2 A - sin 2 B

>> = 1/sin 2 a - sin 2(30)

>> = 1/sin 2a - sin60