Question

If x sin³theta + y cos³theta = sin theta cos theta and x sin theta = y cos theta , prove that x² + y² = 1.


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Answer 1

$→ xsin³∅ + ycos³∅ = sin∅.cos∅$

$→x.sin³∅ + cos²∅.(ycos∅)$

$= sin∅.cos∅$

use xsin∅= ycos∅ in above,

$→ xsin³∅ + cos²∅.xsin∅$

$= sin∅.cos∅$

$→ xsin∅( sin²∅ + cos²∅)$

$= sin∅.cos∅$

$→ xsin∅ = sin∅.cos∅$

$→ x = cos∅$

so,

$→ y = sin∅$

now we know,

$→ sin²∅ + cos²∅ = 1$

put,

$→ sin∅ = y$

$→ cos∅ = x$

hence,

$→x² + y² = 1$

Answer 2

$\begin{gathered}\begin{gathered}\bf \: Given - \begin{cases} &\sf{x \: sin \theta \: = y \: cos \theta} \\ &\sf{x {sin}^{3}\theta \: + y {cos}^{?}\theta \: = sin\theta \:cos\theta \: } \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To \: prove - \begin{cases} &\sf{ {x}^{2} + {y}^{2} = 1 }  \end{cases}\end{gathered}\end{gathered}$

Identity used

$(1). \: \boxed{ \green{ \tt \: {sin}^{2} \theta \: + {cos}^{2} \theta \: = 1}}$

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

Given that

$\rm :\implies \boxed{ \blue{ \bf\:xsin\theta \: = ycos\theta }}\: - - - (1)$

Also,

$\rm :\implies\:x {sin}^{3} \theta \: + y {cos}^{3} \theta \: = sin\theta \:cos\theta \:$

$\rm :\implies\:xsin\theta \: \times {sin}^{2} \theta \: + y {cos}^{ 3} \theta \: = sin\theta \:cos\theta \:$

$\rm :\implies\:ycos\theta \: \times {sin}^{2} \theta \: + y {cos}^{3} \theta \: = sin\theta \:cos\theta \:$

$\rm :\implies\:y \: \: \cancel{cos\theta \:}( {sin}^{2} \theta \: + {cos}^{2} \theta \:) = sin\theta \: \cancel{cos\theta \:}$

$\rm :\implies\: \boxed{ \pink{ \bf \: y \: \tt\: = \: sin\theta \: }} - - - (2)$

On substituting equation (2) in equation (1), we get

$\rm :\implies\:x \: \: \cancel{sin\theta \: }= \cancel{sin\theta} \:cos\theta \:$

$\rm :\implies\: \boxed{ \pink{ \bf \: x \: =  \tt \: cos\theta \:}} - - - (3)$

Now,

Squaring equation (2) and (3) and add, we get

$\rm :\implies\: {x}^{2} + {y}^{2} = {sin}^{2}\theta \: \: + {cos}^{2} \theta \:$

$\rm :\implies\: \boxed{ \green{ \bf \: {x}^{2} + {y}^{2} = 1}}$

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

Additional Information:-

• 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