Suppose - - and - are angles satisfying 0-2- then sin-sin-cos-cos-

Let f(x)=sin x and g(x)=cos x, then f and g are continuous and derivable. Also, sin x ≠ 0 for any x ∈(0, π2) So by Cauchy's mean value theorem, f(β) f(α)g(β) g( ...

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Standard XII

Mathematics

Vn Method

Suppose α, β ...

Question

Suppose $α, β$ and $θ$ are angles satisfying $0 < α < θ < β < \frac{π}{2}$ , then $\frac{sin α - sin β}{cos β - cos α} =$

tan θ

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- tan θ

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cot θ

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- cot θ

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Solution

The correct option is C $cot θ$
Let $f (x) = sin x$ and $g (x) = cos x$ , then $f$ and $g$ are continuous and derivable. Also, $sin x \neq 0$ for any $x \in (0, \frac{π}{2})$
So by Cauchy's mean value theorem, $\frac{f (β) - f (α)}{g (β) - g (α)} = \frac{f^{'} (θ)}{g^{'} (θ)}$

$\Rightarrow \frac{sin β - sin α}{cos β - cos α} = \frac{cos θ}{- sin θ} = - cot θ$
$\Rightarrow \frac{sin α - sin β}{cos β - cos α} = cot θ$

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Similar questions

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Vn Method

Standard XII Mathematics

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Suppose α,β and θ are angles satisfying 0<α<θ<β<π2, then sinα−sinβcosβ−cosα=

Suppose $α, β$ and $θ$ are angles satisfying $0 < α < θ < β < \frac{π}{2}$ , then $\frac{sin α - sin β}{cos β - cos α} =$