Use trigonometric expressions to build an equivalent trigonometric identity with the given expression: (1-cos^2(x))(1+cos^2(x))/cos^2(x)

Question

Use trigonometric expressions to build an equivalent trigonometric identity with the given expression: (1-cos^2(x))(1+cos^2(x))/cos^2(x)

Answer 1

(1-cos^2(x)) = sin²x

 

So the given expression can be written as  sin²x(1+cos²x)/cos²x  which can combine the first part of the numerator & the denominator to become tan²x, so we now have

 

tan²x(1+cos²x) = tan²x + sin²x 

 

since tan²x = sin²x / cos²x & marvelous cancellation occurs when distribution is applied.