How do you verify #csc^2 θ + sec^2 θ = csc^2 θsec^2 θ#?

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Answer 1 · 2018-08-10T03:53:50

#csc^2 θ + sec^2 θ = csc^2 θsec^2 θ#

Since #1+tan^2theta=sec^2theta#:
#csc^2 θ + sec^2 θ = csc^2 θ(1+tan^2theta)#

#csc^2 θ + sec^2 θ = csc^2 θ+csc^2thetatan^2theta#

Since #csctheta=1/sintheta# and #tantheta=sintheta/costheta#:

#csc^2 θ + sec^2 θ = csc^2 θ+1/sin^2theta*sin^2theta/cos^2theta#

#csc^2 θ + sec^2 θ = csc^2 θ+1/cancel(sin^2theta)*cancel(sin^2theta)/cos^2theta#

#csc^2 θ + sec^2 θ = csc^2 θ+1/cos^2theta#

Since #sectheta=1/costheta#:

#csc^2 θ + sec^2 θ = csc^2 θ+sec^2theta sqrt#