Question

Prove that tan theta - cot theta = cosec theta

Answer 1

tan x + cot x = sec x csc x

LHS = sin x / cos x + cos x / sin x

= (sin^2 x + cos^2 x)/(sin x cos x)

= 1/(sin x * cos x)

= sec x csc x = RHS

Another method;   tan(theta)+cot(theta)

=> sin(theta)/cos(theta) + cos((theta)/sin(theta)

=> [sin(theta)*sin(theta)+ cos(theta) * cos(theta)]/sin(theta)cos(theta)

=> [sin^2(theta) + cos^2(theta)]/sin(theta)cos(theta)

Now we know that sin^2(theta) + cos^2(theta) =1 [Identity]

=> 1/sin(theta)cos(theta)

= 1/sin(theta) * 1/cos(theta)

= sec(theta)cosec(theta)