Question

If $7\csc \alpha -3\cot \alpha =7$ prove that $7\cot \alpha -3\csc \alpha =3$

Answer 1

A] Given

$7cosec\theta -3cot\theta =7$

squaring,

$\Rightarrow (7cosec\theta -3cot\theta {)}^2=49$

$\Rightarrow 49cose{c}^2\theta +9co{t}^2\theta -2\times 7\times 3cosec\theta cot\theta =49$

$\Rightarrow 49(1+cot2\theta )+9(cosec2\theta -1)-42cosec\theta cot\theta =49$

$\Rightarrow 49+49cot2\theta +9cosec2\theta -9-42cosec\theta cot\theta =49$

$\Rightarrow \Rightarrow (7cot\theta {)}^2+(3cosec\theta {)}^2-2\times 7\times 3cot\theta cosec\theta =9$

$\Rightarrow (7cot\theta -3cosec\theta {)}^2=9$

$\Rightarrow 7cot\theta -3cosec\theta =3$

Hence proved