Question

evaluate sec 16×cosec 74 -cot 74×tan16 and justify your answer

Answer 1

sec16.cosec(90-16)-cot(90-16).tan74
=sec16.sec16-tan16.tan16
=sec²16-tan²16
=1
It is possible because sec²A-tan²A=1 is true for all values of A which is less than 90°.

Answer 2

We must recall that:

$cosec\theta=sec(90-\theta)\\cot\theta=tan(90-\theta)$

and $cosec^2\theta-cot^2\theta=1$

$sec16 cosec74-cot74 tan16$

So $sec16=cosec74$

and $tan16=cot74$

∴  $cosec74cosec74-cot74cot74$

$=cosec^274-cot^274$

$=1$