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Dear Expert

Kindly assist to solve the following problems:

1. If Cot Theta =2, Find the values of all other Trigonometric Ratios Theta.

2. If 5 Cot Theta = 3, Evaluate 5 Sin Theta - 3 Cos Theta / 5 Sin Theta + 3 Cos Theta.

3. Evaluate Tan²60 + 4 Cos²45 + 8 Cosec²60 / 2 Cosec 30 +3 Sec 60 + 7/3 Cot²30.

4. Evaluate (Cosec²(90 - Theta) - Tan²Theta/ 4(Cos²48 +Cos²42)) - ( 2Tan²30 x Sec²52 x Sin²38 / Cosec² )

5. (Sin Theta -Cos Theta + 1 / Sin Theta + Cos Theta -1)=1 / Sec Theta - Tan Theta.

REQUESTAN URGENT REPLY, PLEASE.

$1. Given cot θ = 2 \tan θ = \frac{1}{2} \sin θ = \frac{1}{\sqrt{5}} \csc θ = \sqrt{5} \cos θ = \frac{2}{\sqrt{5}} \sec θ = \frac{\sqrt{5}}{2} 2. 5 \cot θ = 3 \Rightarrow \cot θ = \frac{3}{5} G i v e n \frac{5 \sin θ - 3 \cos θ}{5 \sin θ + 3 \cos θ} \Rightarrow \frac{5 - 3 \cot θ}{5 + 3 \cot θ} \Rightarrow \frac{5 - 3 (\frac{3}{5})}{5 + 3 (\frac{3}{5})} \Rightarrow \frac{5 - \frac{9}{5}}{5 + \frac{9}{5}} \Rightarrow \frac{25 - 9}{25 + 9} = \frac{16}{34} = \frac{8}{17} 3. \frac{\tan^{2} 60 + 4 \cos^{2} 45 + 8 \csc^{2} 60}{2 \csc 30 + 3 \sec 60 + \frac{7}{3} \cot^{2} 30} \Rightarrow \frac{(\sqrt{3})^{2} + 4 (1)^{2} + 8 {(\frac{2}{\sqrt{3}})}^{2}}{2 (2) + 3 (2) + \frac{7}{3} (\sqrt{3})^{2}} \Rightarrow \frac{3 + 4 + 8 (\frac{4}{3})}{4 + 6 + 7} \Rightarrow \frac{53}{51} 4. Data is missing 5. \frac{\sin θ - \cos θ + 1}{\sin θ + \cos θ - 1} \times \frac{\sin θ + \cos θ + 1}{\sin θ + \cos θ + 1} \Rightarrow \frac{\sin^{2} θ + \sin θ \cos θ + \sin θ - \sin θ \cos θ - \cos^{2} θ - \cos θ + \sin θ + \cos θ + 1}{(\sin θ + \cos θ)^{2} - 1} \Rightarrow \frac{2 \sin^{2} θ + 2 \sin θ}{\sin^{2} θ + \cos^{2} θ + 2 \sin θ \cos θ - 1} \Rightarrow \frac{2 \sin θ (1 + \sin θ)}{2 \sin θ \cos θ} \Rightarrow \frac{1 + \sin θ}{\cos θ} \times \frac{1 - \sin θ}{1 - \sin θ} \Rightarrow \frac{1 - \sin^{2} θ}{\cos θ (1 - \sin θ)} \Rightarrow \frac{\cos^{2} θ}{\cos θ (1 - \sin θ)} \Rightarrow \frac{\cos θ}{1 - \sin θ} R H S \frac{1}{\sec θ - \tan θ} \Rightarrow \frac{1}{\frac{1}{\cos θ} - \frac{\sin θ}{\cos θ}} \Rightarrow \frac{\cos θ}{1 - \sin θ} L H S = R H S ∴ \frac{\sin θ - \cos θ + 1}{\sin θ + \cos θ - 1} = \frac{1}{\sec θ - \tan θ}$

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