Prove Rat: 1) $$ \frac{\sin ^{2} y-\cos ^{2} y}{\cos ^{2} y+\sin y \cdot \cos y}= an y-1 $$ 2) $$ 2 \sin ^{2} 10^{\circ}-4 \cos ^{2} 70+2 \sin ^{2} 80^{\circ}-4 \cos ^{2} 20^{\circ}=-2\left( ight. $$ without $$ \begin{array}{c} ext { calcukator) } \ ext { 3) } \frac{\cos ^{2} x+(1+\sin x)^{2}}{(1+\sin x) \cos x}=\frac{2}{\cos x} \ ext { 4) } \cot ^{2} x- an ^{2} x=\frac{2 \cos ^{2} x-1}{\sin ^{2} x \cos ^{2} x}\end{array} $$

Question

Prove Rat: 1) $$ \frac{\sin ^{2} y-\cos ^{2} y}{\cos ^{2} y+\sin y \cdot \cos y}=	an y-1 $$ 2) $$ 2 \sin ^{2} 10^{\circ}-4 \cos ^{2} 70+2 \sin ^{2} 80^{\circ}-4 \cos ^{2} 20^{\circ}=-2\left(ight. $$ without $$ \begin{array}{c}	ext { calcukator) } \ 	ext { 3) } \frac{\cos ^{2} x+(1+\sin x)^{2}}{(1+\sin x) \cos x}=\frac{2}{\cos x} \ 	ext { 4) } \cot ^{2} x-	an ^{2} x=\frac{2 \cos ^{2} x-1}{\sin ^{2} x \cos ^{2} x}\end{array} $$

UpStudy AI · Accepted Answer

Verify the identity by following steps:
- step0: Verify:
  $$\frac{\cos^{2}\left(x\right)+\left(1+\sin\left(x\right)\right)^{2}}{\left(1+\sin\left(x\right)\right)\cos\left(x\right)}=\frac{2}{\cos\left(x\right)}$$
- step1: Choose a side to work on:
  $$\frac{2}{\cos\left(x\right)}=\frac{2}{\cos\left(x\right)}$$
- step2: Verify the identity:
  $$\textrm{true}$$
Determine whether the expression $$ \frac{\sin ^{2} y-\cos ^{2} y}{\cos ^{2} y+\sin y \cdot \cos y}=\tan y-1 $$ is always true.
Verify the identity by following steps:
- step0: Verify:
  $$\frac{\sin^{2}\left(y\right)-\cos^{2}\left(y\right)}{\cos^{2}\left(y\right)+\sin\left(y\right)\cos\left(y\right)}=\tan\left(y\right)-1$$
- step1: Choose a side to work on:
  $$\tan\left(y\right)-1=\tan\left(y\right)-1$$
- step2: Verify the identity:
  $$\textrm{true}$$
Determine whether the expression $$ 2 \sin ^{2} 10^{\circ}-4 \cos ^{2} 70+2 \sin ^{2} 80^{\circ}-4 \cos ^{2} 20^{\circ}=-2 $$ is always true.
Verify the identity by following steps:
- step0: Verify:
  $$2\sin^{2}\left(10^{\circ}\right)-4\cos^{2}\left(70\right)+2\sin^{2}\left(80^{\circ}\right)-4\cos^{2}\left(20^{\circ}\right)=-2$$
- step1: Choose a side to work on:
  $$-2-2\cos\left(140\right)-2\cos\left(40^{\circ}\right)=-2$$
- step2: Verify the identity:
  $$\textrm{false}$$
Determine whether the expression $$ \cot ^{2} x-\tan ^{2} x=\frac{2 \cos ^{2} x-1}{\sin ^{2} x \cos ^{2} x} $$ is always true.
Verify the identity by following steps:
- step0: Verify:
  $$\cot^{2}\left(x\right)-\tan^{2}\left(x\right)=\frac{2\cos^{2}\left(x\right)-1}{\sin^{2}\left(x\right)\cos^{2}\left(x\right)}$$
- step1: Choose a side to work on:
  $$\frac{1}{\sin^{2}\left(x\right)-\sin^{4}\left(x\right)}-\frac{2}{1-\sin^{2}\left(x\right)}=\frac{2\cos^{2}\left(x\right)-1}{\sin^{2}\left(x\right)\cos^{2}\left(x\right)}$$
- step2: Choose the other side to work on:
  $$\frac{1}{\sin^{2}\left(x\right)-\sin^{4}\left(x\right)}-\frac{2}{1-\sin^{2}\left(x\right)}=\frac{\cos\left(2x\right)}{\sin^{2}\left(x\right)\cos^{2}\left(x\right)}$$
- step3: Verify the identity:
  $$\textrm{false}$$
Let's analyze the results of the proofs for each statement:

1. **Statement 1:** 
   $$
   \frac{\sin ^{2} y - \cos ^{2} y}{\cos ^{2} y + \sin y \cdot \cos y} = \tan y - 1
   $$
   - **Result:** True
   - This equation holds true.

2. **Statement 2:** 
   $$
   2 \sin ^{2} 10^{\circ} - 4 \cos ^{2} 70^{\circ} + 2 \sin ^{2} 80^{\circ} - 4 \cos ^{2} 20^{\circ} = -2
   $$
   - **Result:** False
   - This equation does not hold true.

3. **Statement 3:** 
   $$
   \frac{\cos ^{2} x + (1 + \sin x)^{2}}{(1 + \sin x) \cos x} = \frac{2}{\cos x}
   $$
   - **Result:** True
   - This equation holds true.

4. **Statement 4:** 
   $$
   \cot ^{2} x - \tan ^{2} x = \frac{2 \cos ^{2} x - 1}{\sin ^{2} x \cos ^{2} x}
   $$
   - **Result:** False
   - This equation does not hold true.

### Summary:
- Statements 1 and 3 are true.
- Statements 2 and 4 are false.
- Statement 1 is true. - Statement 2 is false. - Statement 3 is true. - Statement 4 is false.

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