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2 cosec^2 30degree-3sin^2 60degree-3/4tan^2 30degree​

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Prove that [1/(sec2 θ – cos2 θ) + 1/(cosec2 θ – sin2 θ)] sin2 θ.cos2 θ = (1 – sin2 θ.cos2 θ)/(2 + sin2 θ. cos2 θ)

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asked Mar 31, 2019 in Class X Maths by navnit40 (-4,939 points)

Prove that [1/(sec2 θ – cos2 θ) + 1/(cosec2 θ – sin2 θ)] sin2 θ.cos2 θ = (1 – sin2 θ.cos2 θ)/(2 + sin2 θ. cos2 θ)

trigonometry

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answered Mar 31, 2019 by priya12 (-12,631 points)

L.H.S. = [1/(sec2 θ – cos2 θ) + 1/(cosec2 θ – sin2 θ)] sin2 θ.cos2 θ

= [1/(1/cos2 θ – cos2 θ) + 1/(1/sin2 θ – sin2 θ)] sin2 θ.cos2 θ

= [1/(1 – cos4 θ/cos2 θ) + 1/(1 – sin4 θ/sin2 θ)] sin2 θ.cos2 θ

= [cos2 θ/(1 – cos4 θ) + sin2 θ/(1 – sin4 θ)]sin2 θ cos2 θ

= [cos2 θ(1 – sin4 θ) + sin2 θ(1 – cos4 θ)/(1 – cos4 θ)(1 – sin4 θ)] sin2 θ.cos2 θ

= [cos4 θ(1 + sin2 θ) + sin4 θ(1 + cos2 θ)/{sin2 θ cos2 θ(1 + cos2 θ)(1 + sin2 θ)}] sin2 θ cos2 θ

= (cos4 θ + cos4 θ.sin2 θ + sin4 θ + sin4 θ.cos2 θ)/(1 + sin2 θ + cos2 θ + sin2 θ.cos2 θ)

= {(cos2 + sin2 θ)2 – 2 sin2 θ.cos2 θ + sin2 θ.cos2 θ × 1}/2 + sin2 θ.cos2 θ

= ((1)2 – 2 sin2 θ.cos2 θ + sin2 θ.cos2 θ)/2 + sin2 θ.cos2 θ

= (1 – sin2 θ.cos2 θ/(2 + sin2 θ.cos2 θ)

= R.H.S.

Hence proved.

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2 cosec^2 30degree-3sin^2 60degree-3/4tan^2 30degree

Answer 2

$\huge \rm \red{SolOutioN}$

$2 { \cos^{2}60^{\circ}} + \dfrac{ \sec^{2}30^{\circ} }{ \sin^{2}30^{\circ} } + { \cos^{2}45^{\circ}} </p><p>$

Now to find the value of above expression , we have to put :-

Cos 60° = 1/2

sec 30° = 2/√3

sin 30° = 1/2

cos 45° = 1/√2

$⟹ 2 \times ( { \dfrac{1}{2} })^{2} + \dfrac{( {\dfrac{2}{ \sqrt{3} }})^{2} }{ ({ \dfrac{1}{2} })^{2} } + ( {\dfrac{1}{ \sqrt{2} }})^{2}$

$⟹ 2 \times \dfrac{1}{4} + \dfrac{(4 \times 4)}{(3 \times 1)} + \dfrac{1}{2}</p><p> </p><p>$

$⟹ \dfrac{1}{2} + \dfrac{16}{3} + \dfrac{1}{2}$

LCM of 2,3 and 2 = 6

$⟹ \dfrac{3 + 32 + 3}{6}$

$\sf⟹ \dfrac{38}{6}$

$⟹ \dfrac{19}{3} </p><p>$

Answer :- 19/3

_______________________________

Learn more :-

Sin 30° = 1/2

cos 30° = √3/2

tan 30° = 1/√3

Sin 45° = 1/√2

cos 45° = 1/√2

tan 45° = 1

Sin 60° = √3/2

cos 60° = 1/2

tan 60° = √3

Sin 90° = 1

cos 90° = 0

tan 90° = infinite

cosec x = 1/ sin x

sec x = 1/ cosx

cot x = 1/tan x

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