color(blue)(sec(theta)*csc(theta)sec(θ)⋅csc(θ)
We know
color(green)(sec(theta) =1/cos(theta)sec(θ)=1cos(θ)
color(green)(csc(theta) = 1/sin(theta)csc(θ)=1sin(θ)
color(blue)(sec(theta)*csc(theta) = 1/cos(theta)*1/sin(theta)sec(θ)⋅csc(θ)=1cos(θ)⋅1sin(θ)
Note apply the Pythagorean identity
color(green)(cos^2(theta)+sin^2(theta) = 1cos2(θ)+sin2(θ)=1
color(green)(=> sin^2(theta) = (1-cos^2(theta))⇒sin2(θ)=(1−cos2(θ))
color(green)(=>sin(theta) = sqrt(1-cos^2(theta))⇒sin(θ)=√1−cos2(θ)
Our problem would become
sec(theta)*csc(theta) = 1/cos(theta)*1/sin(theta)sec(θ)⋅csc(θ)=1cos(θ)⋅1sin(θ)
color(blue)(sec(theta)*csc(theta) = 1/cos(theta)*1/sqrt(1-cos^2(theta)sec(θ)⋅csc(θ)=1cos(θ)⋅1√1−cos2(θ)
color(blue)(sec(theta)*csc(theta) = 1/(cos(theta)sqrt(1-cos^2(theta))sec(θ)⋅csc(θ)=1cos(θ)√1−cos2(θ)
