Question

if tan tita =1/√7 then the value of cosec²tita-sec²tita/cosec²tita+sec²tita​

Answer 1

$\large\text{\underline{Required}}$

The value of $\dfrac{\csc^{2}\theta-\sec^{2}\theta}{\csc^{2}\theta+\sec^{2}\theta}$.

$\large\text{\underline{Step 1.In terms of one ratio}}$

As we see here, we can express it in terms of $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$.

$\implies\text{(Given)}=\dfrac{\csc^{2}\theta-\sec^{2}\theta}{\csc^{2}\theta+\sec^{2}\theta}$

$\implies\text{(Given)}=\dfrac{\dfrac{1}{\sin^{2}\theta}-\dfrac{1}{\cos^{2}\theta}}{\dfrac{1}{\sin^{2}\theta}+\dfrac{1}{\cos^{2}\theta}}$

We are going to multiply $\sin^{2}\theta$ on both numerator and denominator to express in terms of $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$.

$\implies\text{(Given)}=\dfrac{\dfrac{1}{\sin^{2}\theta}-\dfrac{1}{\cos^{2}\theta}}{\dfrac{1}{\sin^{2}\theta}+\dfrac{1}{\cos^{2}\theta}}$

$\implies\text{(Given)}=\dfrac{1-\tan^{2}\theta}{1+\tan^{2}\theta}$

$\implies\text{(Given)}=\dfrac{1-\dfrac{1}{7}}{1+\dfrac{1}{7}}$

We are going to multiply $7$ on both numerator and denominator.

$\implies\text{(Given)}=\dfrac{7-1}{7+1}$

$\implies\text{(Given)}=\dfrac{3}{4}$

$\large\text{\underline{Conclusion}}$

The required value is $\dfrac{3}{4}$.

Answer 2

Step-by-step explanation:

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