Question

Prove that:
2cos^3 0-cos 0/
sin 0-2sin^3 0= cot 0
​

Answer 1

Step-by-step explanation:

Given Question:-

Prove that:

2cos^3 0-cos 0/sin 0-2sin^3 0= cot 0

Solution:-

LHS:-

(2cos³ 0 - cos 0)/(Sin 0- 2 Sin³ 0)

=>[Cos 0(2 cos² 0-1)]/[Sin 0(1-2 sin² 0)]

=>[Cos 0(2 cos²0-Sin²0 -Cos²0)]/ [ Sin 0 (Sin²0+Cos² 0-2 sin²0)]

=>[Cos 0(Cos²0- Son²0)]/[Sin0(Cos² 0- sin²0)]

Cancelling Cos²0 - sin²0

=>Cos 0/Sin 0

=>Cot 0

=>RHS

LHS=RHS

Answer:-

LHS=RHS

Hence, Proved

Answer 2

Prove that :-

$\tt \:  ⟼ \:\dfrac{2 {cos}^{3}\: \theta - cos\: \theta}{sin\: \theta - 2 {sin}^{3}\: \theta } = cot\: \theta$

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$\begin{gathered}\Large{\bold{\pink{\underline{Formula \: Used \::}}}} \end{gathered}$

$\large \boxed{ \tt \:  \red{ ⟼ (1) \: {sin}^{2} \: \theta + {cos}^{2} \: \theta = 1}}$

$\large \boxed{ \tt \:  \red{ ⟼ (2) \:cot\: \theta \: = \: \dfrac{cos\: \theta}{sin\: \theta} }}$

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$\large\underline\purple{\bold{Solution :- }}$

$\large \boxed{ \tt \:  \red{ ⟼ \:Consider \: LHS \: }}$

$\tt \:  ⟼ \:\dfrac{ 2 {cos}^{3}\: \theta - cos\: \theta}{sin\: \theta - {2sin}^{3}\: \theta }$

$\tt \:  ⟼ \: = \dfrac{cos\: \theta \: ( 2 {cos}^{2} \: \theta - 1)}{sin\: \theta \: (1 - 2 {sin}^{2} \: \theta )}$

$\tt \:  ⟼ = cot\: \theta \times \dfrac{2(1 - {sin}^{2}\: \theta) - 1 }{1 - {2sin}^{2}\: \theta }$

$\tt \:  ⟼ = \: cot\: \theta \times \dfrac{2 - {2sin}^{2} \: \theta - 1}{1 - {2sin}^{2}\: \theta }$

$\tt \:  ⟼ \: = cot\: \theta \times \dfrac{ \cancel{1 - {2sin}^{2} \: \theta}}{\cancel{1 - {2sin}^{2} \: \theta}}$

$\tt \:  ⟼ = \: cot\: \theta$

$\tt \:  ➦ LHS = RHS$

$\large{\boxed{\boxed{\bf{Hence, Proved}}}}$

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$\large \red{\tt \: ⟼ Explore \: \: more } ✍$

Additional Information:-

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

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