Write the trigonometric expression as an algebraic expression containing u and v. cos(tan^-1 u + tan^-1 v)?

cos(tan^-1 u + tan^-1 v)

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    cos(arctan(u) + arctan(v)) =>

    cos(arctan(u))cos(arctan(v)) - sin(arctan(u))sin(arctan(v))

    cos(arctan(u)) =>

    1/sec(arctan(u)) =>

    1/sqrt(1 + tan(arctan(u))^2) =>

    1/sqrt(1 + u^2)

    cos(arctan(v)) =>

    1/sqrt(1 + v^2)

    sin(arctan(u)) =>

    sqrt(1 - cos(arctan(u))^2) =>

    sqrt(1 - 1/sec(arctan(u))^2) =>

    sqrt(1 - 1/(1 + tan(arctan(u))^2)) =>

    sqrt(1 - 1/(1 + u^2)) =>

    sqrt((1 + u^2) - 1) / (1 + u^2)) =>

    sqrt(u^2 / (1 + u^2)) =>

    u / sqrt(1 + u^2)

    sin(arctan(v)) =>

    v / sqrt(1 + v^2)

    cos(arctan(u))cos(arctan(v)) - sin(arctan(u))sin(arctan(v)) =>

    (1/sqrt(1 + u^2)) * (1/sqrt(1 + v^2)) - (u/sqrt(1 + u^2)) * (v/sqrt(1 + v^2)) =>

    (1 - uv) / sqrt((1 + u^2) * (1 + v^2))

  • 4 years ago

    sin(arctan(u) + arctan(v)) = sin(arctan(u))cos(arctan(v)) + cos(arctan(u))sin(arctan(v)) enable z = arctan(u) => tan(z) = u enable x = arctan(v) => tan(x) = v The expression for this reason equals: u/?(u^2 + a million) * a million/?(v^2 + a million) + a million/?(u^2 + a million) * v/?(v^2 + a million) = (u + v)/?((u^2 + a million)(v^2 + a million))

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