Question

if $\large\rm { \big\lgroup \alpha + \beta + \gamma + \delta \big\rgroup = \pi}$ , then
$\large\rm { \displaystyle\sum \cos \alpha \ \cos \beta - \displaystyle\sum \sin \alpha \ \sin \beta = ?}$​

Answer 1

$\large\rm { \displaystyle\sum \cos \alpha \ \cos \beta}$

$\small\rm { = \cos \alpha \ \cos \beta + \cos \beta \ \cos \gamma + \cos \gamma \ \cos \delta + \cos \delta \ \cos \alpha}$

$\large\rm { \displaystyle\sum \sin \alpha \ \sin \beta }$

$\small\rm { = \sin \alpha \ \sin \beta + \sin \beta \ \sin \gamma + \sin \gamma \ \sin \delta + \sin \delta \ \sin \alpha}$

$\large\rm { \displaystyle\sum \cos \alpha \ \cos \beta - \displaystyle\sum \sin \alpha \ \sin \beta }$

$\footnotesize\rm{ = ( \cos \alpha \ \cos \beta + \cos \beta \ \cos \gamma + \cos \gamma \ \cos \delta + \cos \delta \ \cos \alpha) - ( \sin \alpha \ \sin \beta + \sin \beta \ \sin \gamma + \sin \gamma \ \sin \delta + \sin \delta \ \sin \alpha)}$

$\small\rm { = \cos(\alpha + \beta) + \cos(\beta+\gamma) + \cos(\gamma+\delta) + \cos(\delta+\alpha)}$

Thinking Process⚡

we know that $\large\rm { \big\lgroup \alpha + \beta + \gamma + \delta \big\rgroup = \pi}$

So now

$\large\rm { \alpha + \beta = \pi -( \delta + \gamma)}$

$\large\rm { \beta + \gamma = \pi - ( \alpha + \delta)}$

$\small\rm { = - \cos( \gamma + \delta) - \cos(\alpha + \delta) + \cos( \gamma + \delta) + \cos (\alpha + \delta)}$

$\small\rm { = \cancel{ - \cos( \gamma + \delta)} - \cancel{\cos(\alpha + \delta) } + \cancel{ \cos( \gamma + \delta)} + \cancel{\cos (\alpha + \delta)}}$

$\large\rm { = 0}$

Hence, 0 is the required answer.

Answer 2

we know that \large\rm { \big\lgroup \alpha + \beta + \gamma + \delta \big\rgroup = \pi}

⎩

⎪

⎧

α+β+γ+δ

⎭

⎪

⎫

=π

So now

\large\rm { \alpha + \beta = \pi -( \delta + \gamma)}α+β=π−(δ+γ)

\large\rm { \beta + \gamma = \pi - ( \alpha + \delta)}β+γ=π−(α+δ)

\small\rm { = - \cos( \gamma + \delta) - \cos(\alpha + \delta) + \cos( \gamma + \delta) + \cos (\alpha + \delta)}=−cos(γ+δ)−cos(α+δ)+cos(γ+δ)+cos(α+δ)

\small\rm { = \cancel{ - \cos( \gamma + \delta)} - \cancel{\cos(\alpha + \delta) } + \cancel{ \cos( \gamma + \delta)} + \cancel{\cos (\alpha + \delta)}}=

−cos(γ+δ)

−

cos(α+δ)

+

cos(γ+δ)

+

cos(α+δ)

\large\rm { = 0}=0

Hence, 0 is the required answer.