SOLUTION: how can i prove that tan x( cot x + tan x) = sec^2x

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Question 94020This question is from textbook
: how can i prove that tan x( cot x + tan x) = sec^2x This question is from textbook

Answer by Nate(3500) (Show Source):
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tan x( cot x + tan x) = sec^2x
tan x( 1/tan x + tan x) = sec^2x
1 + tan^2 x = sec^2x
1 + sin^2 x / cos^2 x = 1/cos^2x
cos^2 x / cos^2 x + sin^2 x / cos^2 x = 1/cos^2x
( cos^2 x + sin^2 x )/ cos^2x = 1/cos^2x
cos^2 x + sin^2 x = 1
Figure a right triangle:
a^2 + b^2 = c^2
a^2/c^2 + b^2/c^2 = 1
(a/c)^2 + (b/c)^2 = 1
~~~
sin(B) = b/c
cos(B) = a/c
~~~
(a/c)^2 + (b/c)^2 = 1
(cos(B))^2 + (sin(B))^2 = 1
cos^2(B) + sin^2(B) = 1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ x = B
cos^2(B) + sin^2(B) = 1
cos^2(x) + sin^2(x) = 1