( tan ( 90°- theta) ÷ cosec theta +1 ) + ( cosec theta +1 ÷ cot theta) = 2 sec theta prove
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tan(90 - x)/cosecx - 1 + cosecx + 1 / cotx
= cotx/cosecx + 1 + cosecx + 1 / cotx (tan(90 - x) = cotx)
= cot²x + (cosecx + 1)² / cotx(cosecx +1)
= cot²x + cosec²x + 2cosecx + 1 / cotx(cosecx + 1)
= cot²x + cosec²x + cosec²x - cot²x + 2cosecx / cotx(cosecx + 1)
(cosec²x - cot²x = 1)
= 2cosec²x + 2cosecx / cotx(cosecx + 1)
=2cosecx(cosecx + 1) / cotx(cosecx + 1)
= 2cosecx / cotx
= 2 *1/sinx / cosx / sinx
=2 * 1/cosx
= 2secx
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