CBSE Class 10 Answered
if cosec?+cot?=k then show that cos?k²-1/k²+1
Asked by rc1827805 | 23 Dec, 2022, 20:11: PM
![begin mathsize 14px style cos e c space theta space plus space c o t space theta space equals space k end style](https://images.topperlearning.com/topper/tinymce/cache/cfd0e628218033ce0f038673c96fe73a.png)
![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](https://images.topperlearning.com/topper/tinymce/cache/92db7a4b9450bd8092aaf9d5ff058841.png)
![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](https://images.topperlearning.com/topper/tinymce/cache/2ee0c9aa9b3f1dd5b108224135ba2f49.png)
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](https://images.topperlearning.com/topper/tinymce/cache/00f88e24a7c77ee2e98d66c5aafbf861.png)
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](https://images.topperlearning.com/topper/tinymce/cache/78c93879e774606e0c38440396f2fa48.png)
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](https://images.topperlearning.com/topper/tinymce/cache/02e63b95e80e79fe0dbff0af0cd61c68.png)
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](https://images.topperlearning.com/topper/tinymce/cache/82e79dd780823af19eaa5547a0fcdfa9.png)
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](https://images.topperlearning.com/topper/tinymce/cache/ee62aea006db8744b4e0715319f6b671.png)
Answered by Thiyagarajan K | 24 Dec, 2022, 00:25: AM
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