Answer:
1) [tex]\frac{\tan x+1}{\tan x} - \frac{\sec x\cdot \csc x + 1}{\tan x + 1}[/tex] Given
2) [tex]\frac{(\tan x +1)^{2}-\tan x \cdot (\sec x\cdot \csc x +1)}{\tan x\cdot (\tan x + 1)}[/tex] [tex]\frac{a}{b} + \frac{c}{d} = \frac{a\cdot d + b\cdot c}{a\cdot d}[/tex]
3) [tex]\frac{\left(\frac{\sin x}{\cos x} + 1 \right)^{2}-\left(\frac{\sin x}{\cos x} \right)\cdot \left[\left(\frac{1}{\cos x} \right)\cdot \left(\frac{1}{\sin x} \right) + 1\right]}{\left(\frac{\sin x}{\cos x} \right)\cdot \left(\frac{\sin x}{\cos x} +1\right)}[/tex] Identities for tangent, secant and cosecant functions.
4) [tex]\frac{\frac{\sin^{2} x}{\cos^{2} x} + 2\cdot \left(\frac{\sin x}{\cos x} \right) + 1 - \frac{1}{\cos^{2} x} - \frac{\sin x}{\cos x} }{\frac{\sin^{2} x}{\cos^{2} x} +\frac{\sin x}{\cos x} }[/tex] Distributive property/ [tex]\frac{a}{b}\times \frac{c}{d} = \frac{a\cdot c}{b\cdot d}[/tex]
5) [tex]\frac{\frac{\sin ^{2}x +2\cdot \sin x\cdot \cos x +\cos^{2}x -1-\sin x\cdot \cos x}{\cos^{2} x} }{\frac{\sin^{2}x +\sin x \cdot \cos x}{\cos^{2}x} }[/tex] [tex]\frac{a}{b} + \frac{c}{d} = \frac{a\cdot d + b\cdot c}{a\cdot d}[/tex]
6) [tex]\frac{\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}[/tex] [tex]\frac{\frac{a}{b} }{\frac{c}{d} } = \frac{a\cdot d}{b\cdot c}[/tex]
7) [tex]\frac{\sin x\cdot \cos x}{\sin x \cdot (\sin x + \cos x)}[/tex] Fundamental trigonometric identity/Existence of additive inverse/Modulative property/Distributive property
8) [tex]\frac{\cos x}{\sin x + \cos x}[/tex] Associative property/Existence of multiplicative inverse/Modulative property/Result.
Step-by-step explanation:
Now we proceed to demonstrate by algebraic and trigonometric means the following trigonometric identity:
[tex]\frac{\tan x+1}{\tan x} - \frac{\sec x\cdot \csc x + 1}{\tan x + 1} = \frac{\cos x}{\sin x + \cos x}[/tex]
1) [tex]\frac{\tan x+1}{\tan x} - \frac{\sec x\cdot \csc x + 1}{\tan x + 1}[/tex] Given
2) [tex]\frac{(\tan x +1)^{2}-\tan x \cdot (\sec x\cdot \csc x +1)}{\tan x\cdot (\tan x + 1)}[/tex] [tex]\frac{a}{b} + \frac{c}{d} = \frac{a\cdot d + b\cdot c}{a\cdot d}[/tex]
3) [tex]\frac{\left(\frac{\sin x}{\cos x} + 1 \right)^{2}-\left(\frac{\sin x}{\cos x} \right)\cdot \left[\left(\frac{1}{\cos x} \right)\cdot \left(\frac{1}{\sin x} \right) + 1\right]}{\left(\frac{\sin x}{\cos x} \right)\cdot \left(\frac{\sin x}{\cos x} +1\right)}[/tex] Identities for tangent, secant and cosecant functions.
4) [tex]\frac{\frac{\sin^{2} x}{\cos^{2} x} + 2\cdot \left(\frac{\sin x}{\cos x} \right) + 1 - \frac{1}{\cos^{2} x} - \frac{\sin x}{\cos x} }{\frac{\sin^{2} x}{\cos^{2} x} +\frac{\sin x}{\cos x} }[/tex] Distributive property/ [tex]\frac{a}{b}\times \frac{c}{d} = \frac{a\cdot c}{b\cdot d}[/tex]
5) [tex]\frac{\frac{\sin ^{2}x +2\cdot \sin x\cdot \cos x +\cos^{2}x -1-\sin x\cdot \cos x}{\cos^{2} x} }{\frac{\sin^{2}x +\sin x \cdot \cos x}{\cos^{2}x} }[/tex] [tex]\frac{a}{b} + \frac{c}{d} = \frac{a\cdot d + b\cdot c}{a\cdot d}[/tex]
6) [tex]\frac{\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}[/tex] [tex]\frac{\frac{a}{b} }{\frac{c}{d} } = \frac{a\cdot d}{b\cdot c}[/tex]
7) [tex]\frac{\sin x\cdot \cos x}{\sin x \cdot (\sin x + \cos x)}[/tex] Fundamental trigonometric identity/Existence of additive inverse/Modulative property/Distributive property
8) [tex]\frac{\cos x}{\sin x + \cos x}[/tex] Associative property/Existence of multiplicative inverse/Modulative property/Result.
