Maths / Introduction to Trigonometry / Trigonometric Identities

QUESTION

If   $\dpi{120} \fn_jvn \large x\: \: sin^3\Theta +y\: \: \: cos^3\Theta =sin\Theta .cos\Theta\;\; and\: \: \: x\: \: sin\Theta =y$  prove that  $\dpi{120} \fn_jvn \large x^2+y^2=1$ ?

EXPLANATION
Explain TypeExplanation Content
Text

We have,

$x\: \: sin^3\Theta +y\: \: cos^3\Theta =sin\Theta \: \: cos\Theta$

$\Rightarrow \: \: (x\: \: sin\: \: \Theta )sin^2\Theta +(y\: \: cos\: \: \Theta )cos^2\Theta =sin\: \:\Theta \: \: cos\: \Theta$

$\Rightarrow \: \: \: \: x\: sin\: \Theta (sin^2\Theta )+(x\: \: sin\Theta )cos^2\Theta =sin\: \Theta\: \: cos\: \Theta$                       [$\therefore$  x sin $\Theta$  = y cos $\Theta$]

$\Rightarrow \: \: \: \: \: x\: \: sin\Theta (x\: \: \: sin^2\Theta +cos^2\Theta )=sin\Theta \: \: cos\Theta$

$\Rightarrow \: \: \: \: \: x\: \: sin\Theta =sin\Theta \: \: cos\Theta$

$\Rightarrow \: \: \: \: \: x=cos\Theta$

Now,

$x\: \: sin\Theta =y\: \: cos\Theta$

$\Rightarrow \: \: \: \: \: cos \Theta \: sin\Theta =y\: \: cos \Theta$                                                                                               [$\because$ $x=cos\Theta$]

$\Rightarrow \: \: \: \: y=sin\Theta$

Hence,  $x^2+y^2=cos^2\Theta +sin^2\Theta =1.$

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