Can I simplify this any further to include only the cosines

Can I simplify this any further to include only the cosines (hyperbolic and trigonometric)? This is the problem: [sinh(x)*cos(y)]^2 + [cosh(x)*sin(y)]^2.

Solution

We use the properties : cosh²x - sinh²x = 1

                                     sin^2x +cos^2x =1

[sinh(x)*cos(y)]^2 + [cosh(x)*sin(y)]^2.

sinh^2(x)cos^2(y) + cosh^2(x)sin^2(y)

use the prpoerties given above:

(cosh^2x -1)cos^2y + cosh^2(x)(1-cos^2y)

cosh^2xcos^2y - cos^2y + cosh^2x - cosh^2xcos^2y

=cosh^2x - cos^2y