4 Verify the identity tan x sec x cos x 1 sin x 5 Using t

4) Verify the identity tan x + sec x = (cos x) / (1- sin x)

5) Using the Pythagorean Identity sin^2 x + cos ^2 x = 1, derive the other two:

a) tan ^2 x + 1 = sec ^2 x

b) 1 + cot ^2 x = csc ^2 x

Solution

4) tan x + sec x = (sinx + 1)/cosx = (1 - sin^2x)/cosx*(1 - sinx) = cos^2x/[cosx*(1 - sinx)]

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

5) Given sin^2 x + cos ^2 x = 1

Dividing by cos ^2 x, tan^2x + 1 = sec ^2 x

b) Given sin^2 x + cos ^2 x = 1

Dividing by sin ^2 x, 1 + cot^2x = csc^2 x