Question

Prove (1+cot theta / 1-cot theta) + (1-cot theta / 1+cot theta) = 2 / (sin^2 theta - cos^2 theta)

Answer 1

L.H.S

(1+cot@ / 1-cot@) + (1-cot@ / 1+cot@)

= (1 + cot@)^2 + (1 - cot@)^2 / (1^2 - cot^2@)

= (1 + 2cot@ + cot^2@ + 1 - 2cot@ + cot^2@)/ {(sin^2@ - cos^2@)/sin^2@}. {cot@ = cos@/sin@}

= 2(1 + cot^2@) / {(sin^2@ - cos^2@)/sin^2@}

= 2 cosec^2@ / {(sin^2@ - cos^2@)/sin^2@}

= (2 × 1/sin^2@ × sin^2@) / {(sin^2@ - cos^2@)}

= 2/(sin^2@ - cos^2@)

= R.H.S (PROVED)