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sec beta +cosec beta((sin beta +cos beta )^2 -1)

sec can be written in terms of cos and cosec can be written in terms of sin

1/cos beta + 1/sin beta ( sin^2 beta + cos^2 beta + 2sin beta cos beta ) -1

on taking lcm

cos beta + sin beta /cos beta sin beta (1 + 2 sin beta cos beta -1) as cos^2 +sin^2 is one

now 1 - 1 gets cancelled and also cos beta sin beta ets cancelled

so we remain with

2cos beta + 2sin beta

rhs

2m = 2(sin beta + cos beta)

2sin beta + 2cos beta

hens prooved

sinb+cosb=m

secb+cosecb=n => 1/cosb +1/sinb

=> n(m^2-1)=2m

rhs

=> 1/cosb + 1/sinb{(sinb + cosb)^2-1}

(sinb+cosb)/sinbcosb{sin^2b + cos^2b+2sinbcosb - 1}

(sinb+cosb)/sinbcosb(1+2sinbcosb-1)

(sinb+cosb)/sinbcosb(2sinbcosb)

(sinb+cosb)2

2sinb+2cosb

2(sinb+cosb)

hence shown