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Question
If 3 cot θ 4 , show that`((1-tan^2theta))/((1+tan^2theta)) = (cos^2theta - sin^2theta)`
Solution
LHS = `((1-tan^2theta))/((1+tan^2theta))`
=` ((1-1/cot^2theta))/((1+1/cot^2theta))`
=`((cot^2theta-1)/(cot^2theta))/(((cot^2theta+1)/(cot^2theta)))`
=`(cot^2theta-1)/(cot^2theta+1)`
=`((4/3)^2-1)/((4/3)^2 +1)` (𝐴𝑠, 3 cot 𝜃 = 4 𝑜𝑟 cot 𝜃 =`4/3`)
=`(16/9-1)/(16/9+1)`
=`(((16-9)/9))/(((16+9)/9))`
=`((7/9))/((25/9))`
= `7/25`
𝑅𝐻𝑆 = `(cos^2 theta − sin^2 theta)`
=`((cos^2theta - sin^2 theta))/1`
=`(((cos^2theta-sin^2theta)/(sin^2theta)))/((1/(sin^2theta)))`
=`((cos^2theta)/(sin^2theta)-(sin^2theta)/(sin^2theta))/(cosec^2theta)`
=`((cot^2theta-1))/((cot^2theta+1))`
=`([(4/3)^2-1])/([(4/3)^2+1])`
=`((16/9-1/1))/((16/9+1/1))`
=`(((16-9)/9))/(((16+9)/9))`
=`((7/9))/((25/9))`
=`7/25`
Since, LHS = RHS
Hence, verified.
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