Question

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Answer 1

Solution: (Instead of theta I use A,..)

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Given & To find:

15tan²A + 4 sec²A = 23,

(sec²A + cosec²A)² - sin²A = ?
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As we know the 3 identities,

sin²A + cos²A =1,

cosec²A - cot²A = 1,

sec² A - tan²A = 1,..

So,

15 tan²A + 4 sec²A = 23

=> 15 (sec²A - 1) + 4 sec²A = 23

=> -15 + 15sec²A + 4sec²A = 23

=> -15 + 19 sec² A = 23

=>  19 sec²A = 23+ 15

=>  19 sec² A = 38

=> sec²A  =  2,..

=> sec A = $\sqrt{2}$

=> sec A = sec 45°

=> A = 45°

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sec A = $\sqrt{2}$

cosec A = $\sqrt{2}$

sin A = $\frac{1}{ \sqrt{2} }$

and Hence,

(sec A + cosec A)² - sin²A

=> $( \sqrt{2} + \sqrt{2} )^2 - ( \frac{1}{ \sqrt{2} } )^2$

=> $(2 \sqrt{2} )^2 - ( \frac{1}{2} )$

=> $8 - \frac{1}{2}$

=>$7 \frac{1}{2}$

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                                   Hope it Helps !!