Question

Prove that=1/sin10°-√3/cos10°=4

Answer 1

1/ sin 10 - sqrt 3/cos 10 =4 
cos 10 - sqrt 3 sin 10 = 4 cos 10 sin 10 
divide by 2. 
1/2 cos 10 - 1/2 sqrt 3 sin 10= 2 sin 10 cos 10= sin 20. 
put the values 1/2 =sin 30 and 1/2sqrt 3 = cos 30 
sin 30 cos 10 - cos 30 sin 10= sin(30-10)=sin 20 
The left hand side = right hand side. The equation is true

Answer 2

lhs= 1/sin 10 - √3/cos 10

do the lcm

= (1*cos 10 - √3*sin10)/sin10*cos10)
= [(1/2*cos10 -√3/2* sin10)/(1/2 *sin10 cos 10)] here multiplied numerator and denominator with 1/2
=[(sin30cos10 - cos30 sin10)/(2/4* sin10cos10)
=[sin(30-10)/(1/4 *sin 2*10)  { ∵sinAcosB - cosAsinB = sin(A-B)}
=sin20/(1/4 *sin20)       {∵2sinAcosA = sin2A}
after cancellation
=4/1
=4 
=rhs