Question

if cosectheta=t+1/4t(t not included in fraction), then prove that cosectheta + cottheta = 2t or 1/2t

Answer 1

√{(sec β -1)/(secβ +1)}

= √{(1 - cosβ)/(1+cosβ)} (by dividing every term by secβ

= √{(1-cosβ)²/(1-cos²β)} ( by multiplying numerator & denominator by (1-cosβ)

= √{(1- cosβ)² / sin²β }

= (1-cosβ) / sinβ …………………. (1)

Similarly, √{(secβ +1)/(secβ -1)

= √{(1+cosβ) / (1-cosβ) } ( as done above)

= √{(1+cosβ)² / (1-cos²β) }

= √{(1+cosβ)² / sin² β }

= (1+cosβ ) / sinβ ……………. (2)

By adding (1) & (2)

LHS =

= { 1-cosβ + 1+cosβ } / sinβ

= 2/sinβ

= 2 cosec β

= RHS

[ Hence proved]

Answer 2

Underoute {( sec B)/ (Sec B + 1)}

= Underoute {( 1 - cos B)Square/ (1 - Cos Square B)} ( By Multiplying numerator and denominator by (1 - Cos B)

= Unteroute (1 - Cos B) Square /Sin Square B)

(1 - Cos B)/ Sin B.............. (1)

Similarly, Underoute {(Sec B+ 1/ Sec B - 1)

= Underoute {( 1+ Cos B/(1- Cos B)} ( As Done above)
= Underoute {(1+ Cos B) Square/ Sin Square B)
= Underoute {(1+ Cos B)/ Sin B........... (2)
By adding (1) & (2)

LHS =
= ( 1- Cos B + 1+ Cos B)/ Sin B
= 2 Cosec B
= RHS

Hence, Proved.