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CBSE Class 10 Answered

if cosec∅+cot∅=k then show that cos∅k²-1/k²+1
Asked by rc1827805 | 23 Dec, 2022, 08:11: PM
answered-by-expert Expert Answer
begin mathsize 14px style cos e c space theta space plus space c o t space theta space equals space k end style
 
begin mathsize 14px style fraction numerator 1 over denominator sin space theta end fraction plus space fraction numerator cos space theta over denominator sin space theta end fraction space equals space k end style
 
begin mathsize 14px style fraction numerator 1 space plus space cos space theta over denominator sin space theta end fraction space equals space k end style   ...........................(1)
 
If we multiply numerator and denominator by ( 1 - cosθ ) , then we get
 
begin mathsize 14px style fraction numerator sin space theta over denominator 1 space minus space cos space theta end fraction equals space space k
end style ..................................(2)
 
If we multiply eqn.(1) and eqn.(2) , then we get  
 
begin mathsize 14px style fraction numerator 1 space plus space cos space theta over denominator 1 space minus space cos space theta end fraction space equals space k squared end style
 
If we subtract 1 from LHS and RHS , we get after simplification
 
begin mathsize 14px style fraction numerator 2 space cos space theta over denominator 1 space minus space cos space theta end fraction space equals space k squared space minus space 1 end style ............................(3)
 
If we add 1 to LHS and RHS , we get
 
begin mathsize 14px style fraction numerator 2 space over denominator 1 space minus space cos space theta end fraction space equals space k squared space plus space 1 end style  ............................(4)
 
By dividing eqn.(3) by eqn.(4) , we get
 
begin mathsize 14px style cos space theta space equals space fraction numerator k squared minus 1 over denominator k squared plus 1 end fraction end style

 
Answered by Thiyagarajan K | 24 Dec, 2022, 12:25: AM
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