show that... (1+tan²) is equal to sec², so replace that in the denominator since:

sin² + cos² = 1

commute that to

cos² + sin² = 1

and divide by cos²

cos² cos² + sin² cos² = 1 cos²

then, by definitions

1 + tan² = sec²

split the fraktion into its difference use sec² as the common denominator this fraktion needs to be split into its product let's see how much that helped ... we're going to replace a few things by their definitions by definition, 1sec is cos, so swap that if

cos = 1sec

then

cos cos = 1sec 1sec

and

cos² = 1sec²

what else can we replace here? tan is sincos, so swap that out appropriately if

tan = sincos

then

tantan = sincos sincos

and

tan² = sin² cos²

something can cancel out; what do you think? the cos²'s can there's a little trick here you have to try to see it's that cos² is equal to 1-sin² sin² + cos² = 1
sin² + cos² - sin² = 1 - sin²
cos² = 1 - sin² so, replace that from here, there's only a little more algebra;
try grouping the like terms two negative-sine-squares combine to form voltron looks like we're done here we have shown that the original statement is true