Question

Use the Pythagorean identity sin^2 Θ + cos^2 Θ = 1 to derive the other Pythagorean identities, 1 + tan^2 Θ = sec^2 Θ and 1 + cot^2 Θ = csc^2 Θ. Discuss how to remember these identities and other fundamental identities.

Answer 1

1 ) 1 + tan² t = sec² t
1 + sin² t / cos² t = 1 / cos² t
( This is because we know that tan x = sin x / cos x and csc x = 1 / cos x )
Then we will multiply both sides of an equation by cos² t
1 * cos² t + sin² t * cos² t / cos² t = 1
cos² t + sin² t = 1 ( and we know that it is the identity - true )
2 ) 1 + cot ² t = csc² t
1 + cos² t / sin² t = 1 / sin² t / · sin² t ( multiply both sides by sin² t )
sin² t + cos² t = 1 ( true )