Question

please solve this question please do not write wrong solutions I need only right ans
I will mark your answer as branllist if you post right ans

Answer 1

Cot (theta )=1 / root 3

At theta=60 degrees

Put the value in desired equation

{1-cos2(60)}/{2-sin2(60)}

Cos2(60)=1/4

Sin2(60)=3/4

Putting in equation,

{1-1/4}/{2-3/4}

{3/4}{/{5/4}

=3/5

Hence proved...

Answer 2

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Given cotA =1 / √3 ,
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step 1 : solve for sinA
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since we know that

cosecA = √( 1 + cot^2 A )

cosecA = √( 1 + ( 1/√3 )^2 )

cosecA = √(4 / 3 ) = 2 / √3

sinA = 1 / cosecA = 1 / ( 2 / √3 ) = √3 / 2

sinA = √3 / 2

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step2 put value of sinA= √3/ 2 in the given trigonometric function :
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prove: ( 1 - cos^2A )/( 2 - sin^2A) = 3 / 5
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(1 - cos^2A ) / ( 2- sin^2A )

= sin^2A / ( 2 - sin^2A )

= ( sinA )^2 / { 2 - ( sinA )^2 }

= ( √3 / 2 )^2 / { 2 - (√3 /2 )^2 }

= ( 3 / 4 ) /( 2- 3 / 4 )

= 3 / 4 / ( 5 / 4 ) = 3 / 4 × 4 / 5

= 3 / 5 = R. H.S

L.H.S = R.H.S , hence proved

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Note:
(1) 1 - cos^2A = sin^2A

(2) cosecA = √( 1 + cot^2A )
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