​If tan o = 2mn / m² - n² , find the values of the other T-ratios

Dear Student,

Please find below the solution to the asked query:

Given :  tan $\mathrm{\theta }$  = $\frac{2 \mathrm{mn}}{{\mathrm{m}}^2 - {\mathrm{n}}^2}$

We know :  tan $\mathrm{\theta }$ =  $\frac{\mathrm{Opposite}}{\mathrm{Adjacent}}$, So Opposite = 2 m n  and Adjacent = m² - n²

From Pythagoras theorem we know :

Hypotenuse ² = Adjacent² + Opposite²

Hypotenuse² = ( m² - n² )² + ( 2 m n )²

Hypotenuse² = m⁴ + n⁴  - 2  m²  n² + 4 m²  n²

Hypotenuse² = m⁴ + n⁴ + 2 m²  n²

Hypotenuse² = ( m² +  n² )²

Hypotenuse = ( m² +  n² ) 

We know :  Cos $\mathrm{\theta }$ =  $\frac{\mathrm{Adjacent}}{\mathrm{Hypotenuse}}$ , So Cos $\mathrm{\theta }$ = $\frac{\left( {\mathrm{m}}^{2 } - {\mathrm{n}}^{2 }\right)}{\left( {\mathrm{m}}^{2 } + {\mathrm{n}}^{2 }\right)}$

And

Sec $\mathrm{\theta }$ =  $\frac{\mathrm{Hypotenuse}}{\mathrm{Adjacent}}$ , So Sec $\mathrm{\theta }$ = $\frac{\left( {\mathrm{m}}^{2 } + {\mathrm{n}}^{2 }\right)}{\left( {\mathrm{m}}^{2 } - {\mathrm{n}}^{2 }\right)}$
 
And

Sin $\mathrm{\theta }$ =  $\frac{\mathrm{Opposite}}{\mathrm{Hypotenuse}}$ , So Sin $\mathrm{\theta }$  = $\frac{2 \mathrm{m} \mathrm{n}}{\left( {\mathrm{m}}^{2 } + {\mathrm{n}}^{2 }\right)}$

And
Cosec $\mathrm{\theta }$ =  $\frac{\mathrm{Hypotenuse}}{\mathrm{Opposite}}$ , So Cosec $\mathrm{\theta }$  = $\frac{\left( {\mathrm{m}}^{2 } + {\mathrm{n}}^{2 }\right)}{2 \mathrm{m} \mathrm{n}}$


And

Cot $\mathrm{\theta }$ =  $\frac{\mathrm{Adjacent}}{\mathrm{Opposite}}$  Cot $\mathrm{\theta }$ = $\frac{\left( {\mathrm{m}}^2 - {\mathrm{n}}^2 \right)}{2 \mathrm{m} \mathrm{n}}$

Hope this information will clear your doubts about topic.

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