1 Answer

NeedAnswers · Answer 1 · 2022-08-07T05:57:44+0000

Answer:

1)
$(\tan x+1)/(\tan x) - (\sec x\cdot \csc x + 1)/(\tan x + 1)$ Given

2)
$((\tan x +1)^(2)-\tan x \cdot (\sec x\cdot \csc x +1))/(\tan x\cdot (\tan x + 1))$
$(a)/(b) + (c)/(d) = (a\cdot d + b\cdot c)/(a\cdot d)$

3)
$(\left((\sin x)/(\cos x) + 1 \right)^(2)-\left((\sin x)/(\cos x) \right)\cdot \left[\left((1)/(\cos x) \right)\cdot \left((1)/(\sin x) \right) + 1\right])/(\left((\sin x)/(\cos x) \right)\cdot \left((\sin x)/(\cos x) +1\right))$ Identities for tangent, secant and cosecant functions.

4)
$((\sin^(2) x)/(\cos^(2) x) + 2\cdot \left((\sin x)/(\cos x) \right) + 1 - (1)/(\cos^(2) x) - (\sin x)/(\cos x) )/((\sin^(2) x)/(\cos^(2) x) +(\sin x)/(\cos x) )$ Distributive property/
$(a)/(b)* (c)/(d) = (a\cdot c)/(b\cdot d)$

5)
$((\sin ^(2)x +2\cdot \sin x\cdot \cos x +\cos^(2)x -1-\sin x\cdot \cos x)/(\cos^(2) x) )/((\sin^(2)x +\sin x \cdot \cos x)/(\cos^(2)x) )$
$(a)/(b) + (c)/(d) = (a\cdot d + b\cdot c)/(a\cdot d)$

6)
$(\sin^(2)x+2\cdot \sin x \cdot \cos x +\cos^(2)x-1-\sin x\cdot \cos x)/(\sin^(2)x +\sin x \cdot \cos x)$
$((a)/(b) )/((c)/(d) ) = (a\cdot d)/(b\cdot c)$

7)
$(\sin x\cdot \cos x)/(\sin x \cdot (\sin x + \cos x))$ Fundamental trigonometric identity/Existence of additive inverse/Modulative property/Distributive property

8)
$(\cos x)/(\sin x + \cos x)$ Associative property/Existence of multiplicative inverse/Modulative property/Result.

Explanation:

Now we proceed to demonstrate by algebraic and trigonometric means the following trigonometric identity:

$(\tan x+1)/(\tan x) - (\sec x\cdot \csc x + 1)/(\tan x + 1) = (\cos x)/(\sin x + \cos x)$