Question

The value of expression $\frac{(2tan4+3cot4)(2cot4+3tan4)}{24co{t}^{2}8+25}$ is:

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is A 1

$\frac{(2\tan 4+3\cot 4)(2\cot 4+3\tan 4)}{24{\cot }^{2}8+25}$

$=\frac{(\frac{2\sin 4}{\cos 4}+\frac{3\cos 4}{\sin 4})(\frac{2\cos 4}{\sin 4}+\frac{3\sin 4}{\cos 4})}{\frac{24{\cos }^{2}8}{{\sin }^{2}8}+25}$

$=\frac{(2{\sin }^{2}4+3{\cos }^{2}4)(2{\cos }^{2}4+3{\sin }^{2}4)}{{\cos }^4{\sin }^{2}4(24{\cos }^{2}48+25{\sin }^{2}8)}.{\sin }^{2}8$

$=\frac{(2+{\cos }^{2}4)(2+{\sin }^4)}{(24+{\sin }^{2}8)}$

$=\frac{(4+2{\sin }^{2}4+2{\cos }^{2}4+{\sin }^{2}4{\cos }^{2}4)}{(24+{\sin }^{2}8)}.4$

$=\frac{12+4{\sin }^{2}4{\cos }^{2}4}{24+{\sin }^{2}8}$

$=1.$

Hence, the answer is $1.$