Question

Prove that :-
$\tt\dfrac{tanA}{1 - cotA} + \dfrac{cotA}{1 - tanA} = secA + cosecA + 1$​

Answer 1

Formula's Used :-

• tanA = sinA/cosA
• cotA = cosA/sinA
• a³ - b³ = (a - b)(a² + ab + b²)
• sin²A + cos²A = 1
• 1/cosA = secA
• 1/sinA = cosecA

Proof :-

$\: \: \: \: \: \: \: \: \: \:\sf :\implies L.H.S$

$\: \: \: \: \: \: \: \: \pink{\: \:\::\implies \sf \dfrac{tanA}{1 - cotA} + \dfrac{cotA}{1 - tanA}}$

$\: \: \: \: \: \: \: \: \: \:\::\implies \ \sf \dfrac{\bigg(\dfrac{sinA}{cosA}\bigg)}{1 - \bigg(\dfrac{cosA}{sinA}\bigg)} + \dfrac{\bigg(\dfrac{cosA}{sinA}\bigg)}{1 - \bigg(\dfrac{sinA}{cosA}\bigg)}$

$\: \: \: \: \: \: \: \: \: \:\::\implies \ \sf \dfrac{\bigg(\dfrac{sinA}{cosA}\bigg)}{\bigg(\dfrac{sinA - cosA}{sinA}\bigg)} + \dfrac{\bigg(\dfrac{cosA}{sinA}\bigg)}{\bigg(\dfrac{cosA - sinA}{cosA}\bigg)}$

$\: \: \: \: \: \: \: \: \: \:\::\implies \ \sf \bigg(\dfrac{sinA}{cosA} \times \dfrac{sinA}{sinA - cosA}\bigg) + \bigg(\dfrac{cosA}{sinA} \times \dfrac{cosA}{cosA - sinA}\bigg)$

$\: \: \: \: \: \: \: \: \: \:\::\implies \ \sf \dfrac{sinA \times sinA}{cosA \big(sinA - cosA\big)} + \dfrac{cosA \times cosA}{sinA \big(cosA - sinA\big)}$

$\: \: \: \: \: \: \: \: \: \:\::\implies \ \sf \dfrac{sin^2A}{cosA \big(sinA - cosA\big)} + \dfrac{cos^2A}{sinA \big(cosA - sinA\big)}$

$\: \: \: \: \: \: \: \: \: \:\::\implies \ \sf \dfrac{sin^2A}{cosA \big(sinA - cosA\big)} - \dfrac{cos^2A}{sinA \big(sinA - cosA\big)}$

$\: \: \: \: \: \: \: \: \: \:\::\implies \ \sf \dfrac{1}{sinA - cosA} \ \Bigg(\dfrac{sin^2A}{cosA} - \dfrac{cos^2A}{sinA} \Bigg)$

$\: \: \: \: \: \: \: \: \: \:\::\implies \ \sf \dfrac{1}{sinA - cosA} \ \Bigg(\dfrac{sin^3A - cos^3A}{cosAsinA} \Bigg)$

$\: \: \: \: \: \: \: \: \: \:\::\implies \ \sf \dfrac{1}{sinA - cosA} \ \Bigg(\dfrac{\big(sinA - cosA\big) \big(sin^2A + sinAcosA + cos^2A\big)}{cosAsinA} \Bigg)$

$\: \: \: \: \: \: \: \: \: \:\::\implies \ \sf \dfrac{1 + sinAcosA}{cosAsinA}$

$\: \: \: \: \: \: \: \: \: \:\::\implies \ \sf \dfrac{1}{cosAsinA} + \dfrac{sinA cosA}{cosA sinA}$

$\: \: \: \: \: \: \: \: \: \:\::\implies \ \sf \dfrac{1}{cosA} \times \dfrac{1}{sinA} + \dfrac{\cancel{sinA}. \cancel{cosA}}{\cancel{cosA}. \cancel{sinA}}$

$\: \: \: \: \: \: \: \: \: \:\pink{\::\implies \sf secA \times cosecA+ 1}$

$\: \: \: \: \: \: \: \: \: \:\sf: \implies R.H.S$

• Hence, Proved..!

Answer 2

Answer:

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hope this helps you