Question

If $tan\theta =\sqrt{\frac{a}{b}}$ where $a,b$ are positive reals, then the value of $\sin \theta {\sec }^7\theta +\cos \theta co{\sec }^7\theta$ is

• A. $\frac{(a+b{)}^3(a^4+b^4)}{(ab{)}^{7/2}}$
• B. $\frac{(a+b{)}^3(a^4-b^4)}{(ab{)}^{7/2}}$
• C. $\frac{(a+b{)}^3(b^4-a^4)}{(ab{)}^{7/2}}$
• D. None of the above

Answer 1

Answer: (A)

$\frac{(a+b{)}^3(a^4+b^4)}{(ab{)}^{7/2}}$

$\sin \theta .{\sec }^7\theta +\cos \theta .co{\sec }^7\theta$
$=\tan \theta .{\sec }^6\theta +\cot \theta .co{\sec }^6\theta$.
$=\tan \theta (1+{\tan }^2\theta {)}^3+\cot \theta (1+{\cot }^2\theta {)}^3$
$=\sqrt{\frac{a}{b}}(1+\frac{a}{b}{)}^3+\sqrt{\frac{b}{a}}(1+\frac{b}{a}{)}^3$

Upon simplifying, we get
$\frac{(a+b{)}^3(a^4+b^4)}{(ab{)}^{\frac{7}{2}}}$