Write the trigonometric expression cos⁡(sin−1x−cos−1y) as an algebraic expression (that is, without any trigonometric functions). Assume that x and y are positive and in the domain of the given inverse trigonometric function.

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Laaibah Pitt · Accepted Answer

The difference between the two angles based on the cosine terms is given by,  cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β  Substituting the values of alpha and beta in the formula,  cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β  cos⁡(sin−1x−cos−1y)=cos⁡(sin−1x)cos⁡(cos−1y)+sin⁡(sin−1x)sin⁡(cos−1y)  Hence,  sin⁡(sin−1x)=xandcos⁡(cos−1y)=y  On simplifying,  cos⁡(sin−1x)cos⁡(cos−1y)sin⁡(sin−1x)cos⁡(cos−1y)  On applying the Pythagorean identities as shown below,  cos⁡(sin−1x)andsin⁡(cos−1y)  cos⁡(sin−1x)y+xsin⁡(cos−1y)=1−sin2(sin−1x)y+x1−cos2(cos−1y)  On simplification the equation obtained is,  1−sin2(sin−1x)y+x1−cos2(cos−1y)=1−x2y+x1−y2  =y1−x2+x1−y2  Hence,the simplified expression is y1−x2+x1−y2

Jeffrey Jordon · Accepted Answer

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Write the trigonometric expression cos(sin^(-1) x - cos^(-1) y) as an algebraic expression (that is, without any trigonometric functions). Assume that x and y are positive and in the domain of the given inverse trigonometric function.

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