Comparing areas:

  
    
      
        
          
            
              
              
                
                A
                (
                △
                A
                D
                F
                )
                ≥
                A
                (
                
                  sector
                
                
                A
                D
                B
                )
                ≥
                A
                (
                △
                A
                D
                B
                )
              
            
            
              
                ⇒
              
              
                
                
                  
                    1
                    2
                  
                
                ⋅
                tan
                ⁡
                (
                x
                )
                ⋅
                1
                ≥
                
                  
                    x
                    
                      2
                      π
                    
                  
                
                ⋅
                π
                ≥
                
                  
                    1
                    2
                  
                
                ⋅
                sin
                ⁡
                (
                x
                )
                ⋅
                1
              
            
            
              
                ⇒
              
              
                
                
                  
                    
                      sin
                      ⁡
                      (
                      x
                      )
                    
                    
                      cos
                      ⁡
                      (
                      x
                      )
                    
                  
                
                ≥
                x
                ≥
                sin
                ⁡
                (
                x
                )
              
            
            
              
                ⇒
              
              
                
                
                  
                    
                      cos
                      ⁡
                      (
                      x
                      )
                    
                    
                      sin
                      ⁡
                      (
                      x
                      )
                    
                  
                
                ≤
                
                  
                    1
                    x
                  
                
                ≤
                
                  
                    1
                    
                      sin
                      ⁡
                      (
                      x
                      )
                    
                  
                
              
            
            
              
                ⇒
              
              
                
                cos
                ⁡
                (
                x
                )
                ≤
                
                  
                    
                      sin
                      ⁡
                      (
                      x
                      )
                    
                    x
                  
                
                ≤
                1
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}&\,A(\triangle ADF)\geq A({\text{sector}}\,ADB)\geq A(\triangle ADB)\\\Rightarrow &\,{\frac {1}{2}}\cdot \tan(x)\cdot 1\geq {\frac {x}{2\pi }}\cdot \pi \geq {\frac {1}{2}}\cdot \sin(x)\cdot 1\\\Rightarrow &\,{\frac {\sin(x)}{\cos(x)}}\geq x\geq \sin(x)\\\Rightarrow &\,{\frac {\cos(x)}{\sin(x)}}\leq {\frac {1}{x}}\leq {\frac {1}{\sin(x)}}\\\Rightarrow &\,\cos(x)\leq {\frac {\sin(x)}{x}}\leq 1\end{aligned}}}