Question

tan(90-theta)+cot(90-theta) = cosec theta. Sec theta​

Answer 1

Answer:

tan (90° -θ) + cot (90° -θ) = (csc θ)( sec θ)

start with tan (90° -θ) = sin (90° -θ)/cos (90° -θ) = cos θ/ sin θ = cot θ

cot (90° -θ) = cos (90° -θ) / sin (90° -θ) = sin θ/ cos θ = tan θ

so

tan (90° -θ) + cot (90° -θ) = cot θ + tan θ

cot θ + tan θ = cos θ/ sin θ + sin θ/ cos θ = ( cos²θ + sin² θ)/([sin θ][cos θ])

( cos²θ + sin² θ)/([sin θ][cos θ]) = 1/([sin θ][cos θ]) = (1/sin θ)(1/cos θ)

(1/sin θ)(1/cos θ) =( csc θ )( sec θ)

Thus

tan (90° -θ) + cot (90° -θ) = (csc θ)( sec θ)