Question

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Answer 1

Step-by-step explanation:

Given Equation is m = cosA - sinA

Given Equation is n = cosA + sinA.

Numerator:

m² + n² = (cosA - sinA)² + (cosA + sinA)²

            = cos²A + sin²A - 2cosAsinA + cos²A + sin²A + 2cosAsinA

            = 2.

Denominator:

m² - n² = (cosA - sinA)² - (cosA + sinA)²

           = cos²A + sin²A - 2cosAsinA - [cos²A + sin²A + 2cosAsinA]

           = cos²A + sin²A - 2cosAsinA - cos²A - sin²A - 2cosAsinA

           = -4 cosAsinA

Numerator/Denominator:

$=>\frac{2}{-4cosAsinA}$

$=>\frac{-1}{2cosAsinA}$

$=>\frac{-1}{2}*\frac{1}{cosA}*\frac{1}{sinA}$

$=>\frac{-1}{2}secA*cosecA$

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LHS:

$=>-\frac{1}{2} secA * cosecA$

∴ sec²θ - tan²θ = 1 (or) secθ = √1 + tan²θ

∴ cosec²θ - cot²θ = 1 (or) cosecθ = √1+cot²θ

$=>\frac{-1}{2}\sqrt{1+tan^2A} *\sqrt{1+cot^2A}$

$=>\frac{-1}{2}(\sqrt{(1+tan^2A)*(1+cot^2A)})$

$=>\frac{-1}{2}(\sqrt{1+tan^2A+cot^2A+tan^2Acot^2A)}$

$=>\frac{-1}{2}(\sqrt{1+tan^2A+cot^2A+1})$

$=>\frac{-1}{2}\sqrt{tan^2A+cot^2A+2}$

$=>\frac{-1}{2}{\sqrt{(tanA+cotA)^2}}$

$=>\frac{-1}{2}(cotA+tanA)$

$=>\boxed{-\frac{cotA+tanA}{2}}$

RHS.

Hope it helps!

Answer 2

Step-by-step explanation:

m = cosA - sinA

n = cosA + sinA.

m² + n² = (cosA - sinA)² + (cosA + sinA)²

= cos²A + sin²A - 2cosAsinA + cos²A + sin²A + 2cosAsinA

= 2.

m² - n² = (cosA - sinA)² - (cosA + sinA)²

cos²A + sin²A - 2cosAsinA - [cos²A + sin²A + 2cosAsinA]

cos²A + sin²A - 2cosAsinA - cos²A - sin²A - 2cosAsinA

-4 cosAsinA

\frac{2}{-4cosAsinA}

\frac{-1}{2cosAsinA}

\frac{-1}{2}*\frac{1}{cosA}*\frac{1}{sinA}

\frac{-1}{2}secA*cosecA