Question

If 7 cosec - 3 cot =7, prove that 7 cot -3 cosec = ± 3

Answer 1

7cosecθ-3cotθ=7
Squaring both sides,
(7cosecθ-3cotθ)²=49
or, 49cosec²θ-2.7cosecθ.3cotθ+9cot²θ=49
or, 49cosec²θ+9cot²θ-42cosecθcotθ=49
or, 42cosecθcotθ=49cosec²θ+9cot²θ-49
∴, (7cotθ-3cosecθ)²
=49cot²θ-42cosecθcotθ+9cosec²θ
=49cot²θ-49cosec²θ-9cot²θ+49+9cosec²θ
=-49(cosec²θ-cot²θ)+9(cosec²θ-cot²θ)+49
=-49+9+49
[∵, cosec²θ-cot²θ=1]
=9
∴, 7cotθ-3cosecθ=⁺₋ 3 (Proved)