Question

please help me prove the identity​

Answer 1

Answer:

$\frac{1-\tan x }{1+\cot x}$

Step-by-step explanation:

Let $\frac{\sin 2x+\cos 2x-1}{\sin 2x+\cos 2x + 1}$, we proceed to prove the given identity by trigonometric and algebraic means:

1) $\frac{\sin 2x+\cos 2x-1}{\sin 2x+\cos 2x + 1}$ Given

2) $\frac{2\cdot \sin x \cdot \cos x +\cos ^{2}x - \sin^{2}x-1}{2\cdot \sin x \cdot \cos x +\cos ^{2}x - \sin^{2}x+1}$ $\sin 2x = 2\cdot \sin x \cdot \cos x$/$\cos 2x = \cos^{2}x - \sin^{2}x$

3) $\frac{2\cdot \sin x \cdot \cos x -\sin^{2}x-(1-\cos^{2}x)}{2\cdot \sin x\cdot \cos x +\cos^{2}x+(1-\sin^{2}x)}$ Commutative, associative and distributive properties/$-a = (-1)\cdot a$

4) $\frac{2\cdot \sin x \cdot \cos x -2\cdot \sin^{2}x}{2\cdot \sin x \cdot \cos x +2\cdot \cos^{2}x}$ $\sin^{2}x + \cos^{2}x = 1$

5) $\frac{(2\cdot \sin x)\cdot (\cos x-\sin x)}{(2\cdot \cos x)\cdot (\sin x +\cos x)}$ Distributive and associative properties.

6) $\frac{\sin x\cdot (\cos x-\sin x)}{\cos x\cdot (\sin x +\cos x)}$ Existence of multiplicative inverse/Commutative and modulative properties.

7) $\frac{\frac{\cos x -\sin x}{\cos x} }{\frac{\sin x + \cos x}{\sin x} }$ $\frac{\frac{x}{y} }{\frac{w}{z} } = \frac{x\cdot z}{y\cdot w}$

8) $\frac{\frac{\cos x}{\cos x}-\frac{\sin x}{\cos x} }{\frac{\sin x}{\sin x}+\frac{\cos x}{\sin x} }$ $\frac{x+y}{w} = \frac{x}{w} + \frac{y}{w}$

9) $\frac{1-\tan x }{1+\cot x}$ Existence of multiplicative inverse/$\tan x = \frac{\sin x}{\cos x}$/$\cot x = \frac{\cos x}{\sin x}$/Result