Question

what is the value of (tana coseca)^2-(sina seca)^2​

Answer 1

SOLUTION

TO DETERMINE

The value of

$\displaystyle \sf{ {(tan A \: cosec A)}^{2} } - {(sin A \: sec A)}^{2}$

FORMULA TO BE IMPLEMENTED

We are aware of the formula on Trigonometry that

$\displaystyle \sf {sec}^{2} A - {tan}^{2} A = 1$

EVALUATION

Here the given expression is

$\displaystyle \sf{ {(tan A \: cosec A)}^{2} } - {(sin A \: sec A)}^{2}$

We simplify it as below

$\displaystyle \sf{ {(tan A \: cosec A)}^{2} } - {(sin A \: sec A)}^{2}$

$\displaystyle \sf{ = { \bigg( \frac{sin A }{cos A } \: \times \frac{1}{sin A } \bigg)}^{2} } - { \bigg(sin A \times \: \frac{1}{cos A } \bigg)}^{2}$

$\displaystyle \sf{ = { \bigg( \frac{1 }{cos A } \bigg)}^{2} } - { \bigg( \frac{sin A }{cos A } \bigg)}^{2}$

$\displaystyle \sf = {sec}^{2} A - {tan}^{2} A$

$\displaystyle \sf = 1$

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