Prove that 1+tan2 A 1+cot²A sec² A-1

Question :Prove that 1+tan2 A 1+cot²A sec² A-1

The correct answer :We can start with the left-hand side of the given equation:

1 + tan²A

We know that tan²A + 1 = sec²A, so we can substitute that in:

= sec²A – 1 + 1

= sec²A

Now we can substitute sec²A into the right-hand side of the equation:

1 + cot²A * sec²A – 1

= cot²A * sec²A

= cos²A/sin²A * 1/cos²A (since cotA = cosA/sinA and secA = 1/cosA)

= 1/sin²A

= csc²A

Therefore, the left-hand side of the equation simplifies to sec²A, and the right-hand side simplifies to csc²A, which are equal. Thus, we have proven that:

1 + tan²A = 1 + cot²A * sec²A – 1

or, simplifying further:

1 + tan²A = cot²A * sec²A