Maths / Introduction to Trigonometry / Trigonometric Identities

QUESTION
 

If   large x: : sin^3Theta +y: : : cos^3Theta =sinTheta .cosTheta;; and: : : x: : sinTheta =y  prove that  large x^2+y^2=1 ?

EXPLANATION
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Text

We have,

x: : sin^3Theta +y: : cos^3Theta =sinTheta : : cosTheta

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

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

Rightarrow : : : : : x: : sinTheta (x: : : sin^2Theta +cos^2Theta )=sinTheta : : cosTheta

Rightarrow : : : : : x: : sinTheta =sinTheta : : cosTheta

Rightarrow : : : : : x=cosTheta

Now,

        x: : sinTheta =y: : cosTheta

Rightarrow : : : : : cos Theta : sinTheta =y: : cos Theta                                                                                               [because x=cosTheta]

Rightarrow : : : : y=sinTheta

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

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