, prove that : tan²B – sin² B=sin⁴ B sec² B.

We know that,

Or

Let,

Side adjacent to angle B =AB = 12k

Side opposite to angle B =BC = 5k

where, k is any positive integer

Firstly we have to find the value of AC.

So, we can find the value of AC with the help of Pythagoras theorem

⇒ (AB)² + (BC)² = (AC)²

⇒ (12k)² + (5k)² = (AC)²

⇒ (AC)² = 144 k² +25 k²

⇒ (AC)² = 169 k²

⇒ AC =√169 k²

⇒ AC =±13k

But side AC can’t be negative. So, AC = 13k

Now, we will find the sin θ

Side opposite to angle B = BC = 5k

and Hypotenuse = AC = 13k

So,

Now, we know that,

Side adjacent to angle B = AB =12k

Hypotenuse = AC =13k

So,

Now, LHS = tan²B – sin² B

Now, RHS = sin⁴ B sec² B

Now, LHS = RHS

Hence Proved