Question

sin ∅ - 2 sin³ ∅ / 2 cos ³ ∅ - cos ∅
a ) cos ∅
b) sio ∅
c)tan ∅
d) cot ∅​

Answer 1

Answer:

Step-by-step explanation:

We have,

$\tt{\dfrac{sin(\phi)-2\,sin^3(\phi)}{2\,cos^3(\phi)-cos(\phi)}}$

$\tt{=\dfrac{sin(\phi)\big\{1-2\,sin^2(\phi)\big\}}{cos(\phi)\big\{2\,cos^2(\phi)-1\big\}}}$

$\tt{=\dfrac{sin(\phi)\big\{1-2\,(1-cos^2(\phi))\big\}}{cos(\phi)\big\{2\,cos^2(\phi)-1\big\}}}$

$\tt{=\dfrac{sin(\phi)\big\{1-2+2\,cos^2(\phi)\big\}}{cos(\phi)\big\{2\,cos^2(\phi)-1\big\}}}$

$\tt{=\dfrac{sin(\phi)\big\{2\,cos^2(\phi)-1\big\}}{cos(\phi)\big\{2\,cos^2(\phi)-1\big\}}}$

$\tt{=\dfrac{sin(\phi)}{cos(\phi)}}$

$\tt{=tan(\phi)}$

Answer 2

$\large\underline{\sf{Solution-}}$

Given triangle expression is

$\rm :\longmapsto\:\dfrac{sin\phi - {2sin}^{3}\phi}{ {2cos}^{3}\phi - cos\phi }$

$\rm \:  =  \: \dfrac{sin\phi (1 - {2sin}^{2} \phi )}{cos\phi ( {2cos}^{2}\phi - 1)}$

We know,

$\rm :\longmapsto\:\boxed{\tt{ {sin}^{2} x + {cos}^{2}x = 1 \: }}$

So, using this identity, the above expression can be rewritten as

$\rm \:  =  \: \dfrac{sin\phi ( {sin}^{2}\phi + {cos}^{2}\phi - {2sin}^{2} \phi )}{cos\phi ( {2cos}^{2}\phi - [{sin}^{2}\phi + {cos}^{2}\phi ])}$

$\rm \:  =  \: \dfrac{sin\phi ( {cos}^{2}\phi - {sin}^{2} \phi )}{cos\phi ( {2cos}^{2}\phi - {sin}^{2}\phi - {cos}^{2}\phi )}$

$\rm \:  =  \: \dfrac{sin\phi ( {cos}^{2}\phi - {sin}^{2} \phi )}{cos\phi ( {cos}^{2}\phi - {sin}^{2}\phi )}$

$\rm \:  =  \: \dfrac{sin\phi }{cos\phi }$

$\rm \:  =  \: tan\phi$

Hence,

$\rm :\longmapsto\:\boxed{\tt{ \dfrac{sin\phi - {2sin}^{3}\phi}{ {2cos}^{3}\phi - cos\phi } = tan\phi }}$

So, Option (c) is Correct

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Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1