Question

Without using trigonometric tables, prove that:

cosec 39° cos 51° + tan 21° cot 69° – sec²21° = 0

Answer 1

Given: cosec 39° · cos 51° + tan 21° · cot 69° – sec² 21°

To Prove: cosec 39° cos 51° + tan 21° cot 69° – sec² 21° = 0

Proof [Step-wise]:

1. Note complementary-angle identities:

cos(51°) = sin(39°) and cot(69°)

= cot(90° – 21°) 

= tan 21°

2. Evaluate the first product:

`"cosec"  39^circ · cos 51^circ = (1/sin 39^circ) · cos 51^circ` 

= `(1/sin 39^circ) · sin 39^circ` 

= 1

3. Evaluate the second product:

tan 21° · cot 69° = tan 21° · cot(90° – 21°) 

= tan 21° · tan 21°

= tan² 21°

4. Substitute into the expression:

1 + tan² 21° – sec² 21°

5. Use the Pythagorean identity 1 + tan²θ = sec²θ (with θ = 21°): 

1 + tan² 21° = sec² 21°

So 1 + tan² 21° − sec² 21°

= sec² 21° − sec² 21° 

= 0

cosec 39° cos 51° + tan 21° cot 69° – sec² 21° = 0.

Hence proved.