expand left side
recall that tan^2 = sec^2 - 1
done
prove: (tan x + sec x)^2 = 2sec^2 x + 2tan x sec x - 1
3 answers
left side
(sin/cos + 1/cos)^2
(1/cos^2)(sin^2 + 2 sin + 1)
right side
2/cos^2 + 2 sin/cos^2 - cos^2/cos^2)
(1/cos^2)( 2 + 2 sin -(1-sin^2)
(1/cos^2)(2 + 2 sin -1 + sin^2)
(1/cos^2)( sin^2 + 2 sin + 1) remarkable
(sin/cos + 1/cos)^2
(1/cos^2)(sin^2 + 2 sin + 1)
right side
2/cos^2 + 2 sin/cos^2 - cos^2/cos^2)
(1/cos^2)( 2 + 2 sin -(1-sin^2)
(1/cos^2)(2 + 2 sin -1 + sin^2)
(1/cos^2)( sin^2 + 2 sin + 1) remarkable
Much more fun to do it the hard way