Question

Tan theta +4cot theta=4 then find 1)tan^2 theta+cot^2 theta 2)cosec^2 theta+sec^2 theta

Answer 1

tanθ+4cotθ=4
or, tanθ+4/tanθ=4
or, tan²θ+4=4tanθ
or, tan²θ-4tanθ+4=0
or, (tanθ-2)²=0
or, tanθ-2=0
or, tanθ=2
∴, cotθ=1/tanθ=1/2
1) tan²θ+cot²θ
=2²+(1/2)²
=4+1/4
=(16+1)/4
=17/4 
2) tanθ=2
We know that, sec²θ-tan²θ=1
or, sec²θ=1+tan²θ
or, sec²θ=1+2²
or, sec²θ=5
Also, cosec²θ-cot²θ=1
or, cosec²θ=1+cot²θ
or, cosec²θ=1+(1/2)²
or, cosec²θ=5/4
∴, cosec²θ+sec²θ
=5/4+5
=25/4