Question

Prove that
Sin2x+cos3x+cosec4xsin2x=4

Answer 1

From basic definitions and the Pythagorean Theorem
cos2(x)+sin2(x)=1
or
cos2(x)=1−sin2(x)

First consider
(sin2(x)−sin4(x)

=(sin2(x))⋅(1−sin2(x))

=sin2(x)cos2(x)usedbelow

So
cos3(x)sin2(x)

=(cos(x))⋅⎡⎢ ⎢⎣(cos2(x)sin2(x))asabove⎤⎥ ⎥⎦

=(cos(x))⋅(sin2(x)−sin4(x))

=(sin2(x)−sin4(x))cos(x)