Question

If tanA +4cotA=4,find the value of (1)tan2A+cot2A and (2)cosec2A+sec2A

Answer 1

tanA+4cotA=4
or, tanA+4/tanA=4
or, (tan²A+4)/tanA=4
or, tan²A+4=4tanA
or, tan²A-2.tanA.2+2²=0
or, (tanA-2)²=0
or, tanA-2=0
or, tanA=2
(1) tan²A+cot²A
=tan²A+1/tan²A
=2²+1/2²
=4+1/4
=(16+1)/4
=17/4
(2) cosec²A+sec²A
=(1+cot²A)+(1+tan²A) [∵, cose²Ф-cot²Ф=1 and sec²Ф-tan²Ф=1]
=2+1/tan²A+tan²A
=2+1/2²+2²
=2+1/4+4
=6+1/4
=(24+1)/4
=25/4