Question

If $\tan \alpha +\cot \alpha =a$ then the value of ${\tan }^4\alpha +{\cot }^4\alpha =$

Answer 1

We have,

$\tan \alpha +\cot \alpha =a$ …… (1)

${\tan }^4\alpha +{\cot }^4\alpha =?$

Squaring both side and we get,

${(\tan \alpha +\cot \alpha )}^2=a^2$

${\tan }^2\alpha +{\cot }^2\alpha +2\tan \alpha \cot \alpha =a^2$

${\tan }^2\alpha +{\cot }^2\alpha +2=a^2$

${\tan }^2\alpha +{\cot }^2\alpha =a^2-2$

Again, squaring both side and we get,

${({\tan }^2\alpha +{\cot }^2\alpha )}^2={(a^2-2)}^2$

${\tan }^4\alpha +{\cot }^4\alpha +2{\tan }^2\alpha {\cot }^2\alpha =a^4+4-4a$

${\tan }^4\alpha +{\cot }^4\alpha +2=a^4+4-4a$

${\tan }^4\alpha +{\cot }^4\alpha =a^4+4-4a-2$

${\tan }^4\alpha +{\cot }^4\alpha =a^4-4a-2$

Hence, this is the answer.