CBSE Class 12-science Answered
if sin x+2cos x=1 then find Tha value of 2sin x - cosx
Asked by gopalkumarrai87 | 11 Oct, 2023, 11:02: AM
Let t = tan(x/2)
![begin mathsize 14px style sin left parenthesis x right parenthesis space equals space fraction numerator 2 space tan left parenthesis x divided by 2 right parenthesis over denominator 1 plus space tan squared left parenthesis x divided by 2 right parenthesis space end fraction space equals space fraction numerator 2 space t over denominator 1 space plus space t squared end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/cdf3a02e4d456a91dea44833e5367ddd.png)
![begin mathsize 14px style cos left parenthesis x right parenthesis space equals fraction numerator 1 space minus space tan squared left parenthesis x divided by 2 right parenthesis over denominator 1 space plus space tan squared left parenthesis x divided by 2 right parenthesis end fraction space equals space fraction numerator 1 space minus space t squared over denominator 1 plus t squared end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/90d4120bcbb11b0b35856403c572dcaf.png)
![begin mathsize 14px style sin space left parenthesis x right parenthesis space plus space 2 space cos space left parenthesis x right parenthesis space equals space 1 end style](https://images.topperlearning.com/topper/tinymce/cache/14da676b8ab8914ddf712e285bd8eee4.png)
![begin mathsize 14px style fraction numerator 2 space t over denominator 1 space plus space t squared end fraction space plus space space 2 cross times fraction numerator 1 space minus space t squared over denominator 1 plus t squared end fraction space equals space 1 end style](https://images.topperlearning.com/topper/tinymce/cache/c7ed795cfacedf5f739e2680ee9e2970.png)
![begin mathsize 14px style 2 space t space plus space 2 space minus space 2 space t squared space equals space 1 space plus space t squared end style](https://images.topperlearning.com/topper/tinymce/cache/55d766b92a2b87aae2e52def97e179f0.png)
![begin mathsize 14px style 3 space t squared minus space 2 t space minus space 1 space equals space 0 end style](https://images.topperlearning.com/topper/tinymce/cache/97a437b72374e4ffec0eea9fc22ca4a0.png)
![begin mathsize 14px style left parenthesis space t space minus 1 space right parenthesis space left parenthesis space 3 space t space plus space 1 space right parenthesis space equals space 0 end style](https://images.topperlearning.com/topper/tinymce/cache/ddd1ac663f8687449bd446de9aa69c3c.png)
Hence t = 1 or t = -1/3
-------------------------------
if t = tan(x/2) = 1 , then x =
/2
![begin mathsize 14px style straight pi end style](https://images.topperlearning.com/topper/tinymce/cache/df87b0d41373b1aae9f4b0b7068f8599.png)
2 sin(x) - cosx = 2 - 0 = 2
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if t = -1/3 , then
![begin mathsize 14px style 2 space sin left parenthesis x right parenthesis space minus space cos space left parenthesis x right parenthesis space equals space 2 space cross times fraction numerator 2 t over denominator 1 plus t squared end fraction minus space fraction numerator 1 space minus space t squared over denominator 1 plus t squared end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/78942f14d5a1740a217499989c03af31.png)
![begin mathsize 14px style R H S space equals space fraction numerator 9 over denominator 10 space end fraction cross times open square brackets negative 4 over 3 minus space 1 space plus 1 over 9 close square brackets space equals space 9 over 10 open square brackets fraction numerator negative 20 over denominator 9 end fraction close square brackets space equals space minus 2 end style](https://images.topperlearning.com/topper/tinymce/cache/3e81e53bb49fad4ad2c06ec672ac6eee.png)
2 sin(x) - cosx = -2
Answered by Thiyagarajan K | 11 Oct, 2023, 23:00: PM
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