Solve this follows e R and a s b'} is neither reflexive nor symmetric nor transitive 22 Prove - Biology - Biodiversity and Conservation

Question

Solve this follows

Priyanka Sahu · Accepted Answer

Dear Student, Taking L H S , $\cos [\tan^{- 1} {\sin (\cot^{- 1} x)}] Let \cot^{- 1} x = θ \Rightarrow x = \cot θ ∴ \cos [\tan^{- 1} {\sin (\cot^{- 1} x)}] = \cos [\tan^{- 1} {\sin θ}] (1) Now, \cot θ = x Also, 1 + \cot^{2} θ = {cosec}^{2} θ \Rightarrow 1 + x^{2} = {cosec}^{2} θ \Rightarrow cosec θ = \sqrt{1 + x^{2}} \Rightarrow \sin θ = \frac{1}{\sqrt{1 + x^{2}}} [∵ \sin θ = \frac{1}{cosec θ}]$ ∴ From (1), we get – $\cos [\tan^{- 1} {\sin (\cot^{- 1} x)}] = \cos [\tan^{- 1} (\frac{1}{\sqrt{1 + x^{2}}})] (3) Let \tan^{- 1} (\frac{1}{\sqrt{1 + x^{2}}}) = y \Rightarrow \frac{1}{\sqrt{1 + x^{2}}} = \tan y (4)$ From (3), $\cos [\tan^{- 1} {\sin (\cot^{- 1} x)}] = \cos y (5)$ Our aim is to find the value of cos y $Since, \tan y = \frac{1}{\sqrt{1 + x^{2}}} and \sec^{2} y = 1 + \tan^{2} y = 1 + \frac{1}{1 + x^{2}} \Rightarrow \sec^{2} y = \frac{2 + x^{2}}{1 + x^{2}} \Rightarrow \cos^{2} y = \frac{1 + x^{2}}{2 + x^{2}} \Rightarrow \cos y = \sqrt{\frac{1 + x^{2}}{2 + x^{2}}}$ ∴ From (5), $\cos [\tan^{- 1} {\sin (\cot^{- 1} x)}] = \sqrt{\frac{1 + x^{2}}{2 + x^{2}}}$ Hence, RHS = LHS Regards

Solve this follows Solve this follows e R and a s b'} is neither reflexive nor symmetric nor transitive. 22. Prove that cos tan-v sin (cot-I x)} =

Solve this follows

Solve this follows e R and a s b'} is neither reflexive nor symmetric nor transitive. 22. Prove that cos tan-v sin (cot-I x)} =