How do I simplify with remaining quotients 1 cot2 Theta1 t

How do I simplify with remaining quotients 1 + cot^2 Theta/1 + tan^2 Theta

Solution

we have given [1+cot²(theta)]/[1+tan²(theta)]

[1+cot²(theta)]/[1+tan²(theta)]=[1+cos²(theta)/sin²(theta)]/[1+sin²(theta)/cos²(theta)]

=[(sin²(theta)+cos²(theta))/sin²(theta)]/[(cos²(theta)+sin²(theta))/cos²(theta)]

=[(sin²(theta)+cos²(theta))/sin²(theta)]*[cos²(theta)/(cos²(theta)+sin²(theta))]

=[cos²(theta)]/[sin²(theta)]

=cot²(theta) since sin²(theta)+cos²(theta)=1

[1+cot²(theta)]/[1+tan²(theta)] =cot²(theta)