Question

Tan^2 theta - cot^2 theta = sec^2 theta (1-cot^2 theta)​

Answer 1

Based on two Pythagorean Identities:

• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ

sec²θ(1 - cot²θ)

= sec²θ - sec²θ · cot²θ

`=(1)/(cos ^(2) \theta) -(1)/(cos ^(2)\theta )*(cos ^(2)\theta)/(sin^(2) \theta)\\\\=(1)/(cos ^(2)\theta )-(1)/(sin^(2)\theta )\\\\=sec^(2)\theta-csc^(2)\theta\\\\=1+tan^(2)\theta-(1+cot^(2)\theta)\\\\=1+tan^(2)\theta-1-cot^(2)\theta\\\\=tan^(2)\theta-cot^(2)\theta`