Question

Prove the following:
Tan²θ + cos²θ = sec²θ × cosec²θ - 2
(Class 10 | Mathematics | Introduction to Trigonometry)

Answer 1

Answer:

To prove ----->

( Sec²θ - 1 ) ( Cosec²θ - 1 ) = 1

Proof-----> We know that ,

1 + tan²θ = Sec²θ

=> tan²θ = Sec²θ - 1

We know that ,

1 + Cot²θ = Cosec²θ

=> Cot²θ = Cosec²θ - 1

LHS = ( Sec²θ - 1 ) ( Cosec²θ - 1 )

Putting Sec²θ - 1 = tan²θ and Cosec²θ - 1 = Cot²θ , we get ,

= ( tan²θ ) ( Cot²θ )

= ( tan²θ ) ( 1 / tan²θ )

= 1 = RHS

Additional information------->

1) Sin²θ + Cos²θ = 1

2) Sin²θ = 1 - Cos²θ

3) Cos²θ = 1 - Sin²θ

4) 1 + tan²θ = Sec²θ

5) tan²θ = Sec²θ - 1

6) Sec²θ - tan²θ = 1

7) 1 + Cot²θ = Cosec²θ

8) Cot²θ = Cosec²θ - 1

9) Cosec²θ - Cot²θ = 1

Answer 2

Explanation:

to prove tan²A + cot²A = sec²A × cosec²A - 2

RHS : (1 + tan²A)(1 + cot²A) - 2

=> 1 + cot²A + tan²A + tan²A * cot²A - 2

=> 1 + cot²A + tan²A + 1 - 2

=> cot²A + tan²A => RHS

hence proved