Question

I need help with this question. Verify identity?!

Answer 1

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)`

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.