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

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.