sec^2x-1=tan^2x?

Question

how do you prove that these two are equal?

Answer 1

sec^2(x)-1
=1/cos^2(x) - 1
= ( 1 - cos^2(x) ) /cos^2(x)
= sin^2(x) / cos^2(x)
= tan^2(x)

Answer 2

1) First, let U = tan^2 + 1 - sec^2; and note that your equation is true if U equals zero

2) Change sec and tan into sin and cos and all will be revealed
U = (sin^2 / cos^2) + 1 - (1/cos^2)

3) Multiply the RHS by (cos^2 / cos^2) and you're done
U = (sin^2 + cos^2 -1 ) / cos^2 = (1 - 1)/ cos^2 = 0

Answer 3

work on the left side and use a common denominator of cos²x:

sec²x - 1
= 1/cos²x - cos²x/cos²x
= (1-cos²x)/cos²x
= sin²x / cos²x
=tan²x

Answer 4

x and y axes, unit circle, angle Θ in quadrant 1, horizontal side is cos Θ, vertical side is sin Θ, and by Pythagorean Theorem we have

sin² Θ + cos² Θ = 1

let's replace Θ with x, and divide both sides by cos² x:

sin² x ..... cos² x ....... 1
-------- + ---------- = ----------
cos² x .... cos² x ......cos² x

tan² x + 1 = sec² x
tan² x = sec² x - 1