Prove each of the following identities.

(e) (2 - sec^2 x)/(sec^2 x + 2tan x) = (cos x - sin x)/(cos x + sin x)

(f) cot^2 x - cot^2 y = (cos^2 x - cos^2 y)/(sin^2 x * sin^2 y)​

The given trigonometry proof is given below

(a)[tex]\frac{1}{1+\sin (\theta)}+\frac{1}{1-\sin (\theta)}=2 \sec ^2(\theta)[/tex]

Manipulating left side

[tex]$$\begin{aligned}&\frac{1}{1+\sin (\theta)}+\frac{1}{1-\sin (\theta)} \\&\text { Simplify } \frac{1}{1+\sin (\theta)}+\frac{1}{1-\sin (\theta)}: \frac{2}{(\sin (\theta)+1)(-\sin (\theta)+1)} \\&=\frac{2}{(1+\sin (\theta))(1-\sin (\theta))}\end{aligned}$$[/tex]

Rewrite using trig identities

[tex]=\frac{2}{\cos ^2(\theta)}[/tex]

Use the basic trigonometric identity: [tex]\frac{1}{\cos (x)}=\sec (x)[/tex]

[tex]$$=2 \sec ^2(\theta)$$[/tex]

(b)[tex]\frac{\cos (x)}{1+\sin (x)}=\sec (x)-\tan (x)[/tex]

Manipulating right side

[tex]$$\sec (x)-\tan (x)$$[/tex]

Express with sin, cos

[tex]=\frac{1-\sin (x)}{\cos (x)}[/tex]

Rewrite using trig identities

[tex]=\frac{\cos (x)}{1+\sin (x)}[/tex]

(c)Manipulating right side[tex]$\tan (x)-\cot (x)$[/tex]

Express with sin, [tex]$\cos$[/tex]

[tex]$$\begin{aligned}&=\frac{-\cos ^2(x)+\sin ^2(x)}{\cos (x) \sin (x)} \\&\sin ^2(x)=1-\cos ^2(x) \\&=\frac{1-\cos ^2(x)-\cos ^2(x)}{\cos (x) \sin (x)} \\&=\frac{1-2 \cos ^2(x)}{\sin (x) \cos (x)}\end{aligned}$$[/tex]

(d) [tex]\frac{\sin (x)}{1-\cot (x)}+\frac{\cos (x)}{1-\tan (x)}=\sin (x)+\cos (x)[/tex]

Manipulating left side

[tex]\frac{\sin (x)}{1-\cot (x)}+\frac{\cos (x)}{1-\tan (x)}[/tex]

Express with sin, cos

[tex]=\cos (x)+\sin (x)[/tex]

(e)Manipulating left side

[tex]\frac{2-\sec ^2(x)}{\sec ^2(x)+2 \tan (x)}[/tex]

Express with sin, cos

[tex]=\frac{-1+2 \cos ^2(x)}{1+2 \cos (x) \sin (x)}$$[/tex]

Rewrite using trig identities

[tex]=\frac{\cos (x)-\sin (x)}{\cos (x)+\sin (x)}$$[/tex]

(f)Manipulating left side [tex]$\cot ^2(x)-\cot ^2(y)$[/tex]

Express with sin, cos

[tex]=\frac{\cos ^2(x) \sin ^2(y)-\cos ^2(y) \sin ^2(x)}{\sin ^2(x) \sin ^2(y)}$$[/tex]

Rewrite using trig identities

[tex]=\frac{\cos ^2(x)-\cos ^2(y)}{\sin ^2(x) \sin ^2(y)}$$[/tex]

Trigonometry is the field of mathematics that deals with certain angles' functions and how to use such functions in computations. There are six popular trigonometric functions for an angle. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant are their respective names and acronyms (csc).

The area of mathematics known as trigonometry examines the link between the ratios of a right-angled triangle's sides to its angles. Trigonometric ratios, such as sine, cosine, tangent, cotangent, secant, and cosecant, are employed to analyze this connection.

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