Proof of Sin 70/cos 20 cosec20/sec 70 -2cos 70 cosec 20 = 0:

Step 1: Simplify the given expression using trigonometric identities:

Sin 70/cos 20 = (Sin 70/sin 70 cos 20) = (1/cos 20)

cosec 20/sec 70 = (1/sin 20 cos 70) = (1/((1/2)(sin 70))) = (2/sin 70)

-2cos 70 cosec 20 = (-2cos 70/sin 20)

Substituting these values in the given expression, we get:

(1/cos 20) * (2/sin 70) - (2cos 70/sin 20) = (2/cos 20 sin 70) - (2cos 70/sin 20)

Step 2: Use the identity cos A sin B = (1/2)(sin(A + B) - sin(A - B)) to simplify the expression further:

cos 20 sin 70 = (1/2)(sin(20 + 70) - sin(70 - 20)) = (1/2)(sin 90 - sin 50) = (1/2)(1 - sin 50)

Substituting this value in the expression, we get:

(2/cos 20 sin 70) - (2cos 70/sin 20) = (4/(1 - sin 50) cos 20) - (2cos 70/sin 20)

Step 3: Use the identity cos A = sin (90 - A) to simplify the expression further:

cos 70 = sin (90 - 70) = sin 20

Substituting this value in the expression, we get:

(4/(1 - sin 50) cos 20) - (2cos 70/sin 20) = (4/(1 - sin 50) cos 20) - (2sin 20/sin 20) = (4/(1 - sin 50) cos 20) - 2

Step 4: Use the identity 1 - sin^2 A = cos^2 A to simplify the expression further:

1 - sin^2 50 = cos^2 50

cos 50 = sqrt(1 - sin^2 50) = sqrt(cos^2 40)

Substituting this value in the expression, we get:

(4/(cos^2 40 - cos^2 50) cos 20) - 2 = (4cos 20)/(cos 40 + cos 50) - 2

Step 5: Use the identity cos A + cos B = 2cos((A + B)/2)cos((A - B)/2) to simplify the expression further:

cos 40 + cos 50 = 2cos(45)cos(5/2) = sqrt(2)(cos(5/2))

Substituting this value in the expression, we get:

(4cos 20)/(sqrt(2)(cos(5/2))) - 2 = (2sqrt(2)cos