Prove the identity Cosx1sinx sec x tan x Note that each St

Prove the identity. Cos(-x)/1-sin(-x) = sec x - tan x Note that each Statement must be based on a select the corresponding question mark.

Solution

cos(-x)/(1-sin(-x))=secx-tanx

Simplifying left side

Cos(-x)/(1-sin(-x))

cos x/(1+sin x)

multiplying both numerator and denominator by 1-sin x

cos x(1-sin x)/(1+sin x)(1-sin x)

cos x(1-sin x)/(1-sin²x)

cos x(1-sin x)/cos²x

(1-sin x)/cos x

Simplifying right side

sec x - tan x

1/cos x - sin x/cos x

(1 - sinx) / cos x

Left side = Right side

Hence cos(-x)/(1-sin(-x))=sec x - tan x