A vertical rod is fixed in a horizontal rectangular field ABCD.The angular elevations of its top A ,B,C and D areα , β, γ and δ

respectively.Show that cot ² α - cot² β = cot ² δ - cot ² γ

Question

A vertical rod is fixed in a horizontal rectangular field ABCD
The angular elevations of its top A ,B,C and D
areÎ± , Î²,
Î³ and Î´
respectively Show that cot
2 Î± -
cot2 Î² =
cot 2 Î´
- cot 2
Î³

Ajanta Trivedi · Accepted Answer

let l and b be the length and breadth of the rectangular field ABCD PQ is the vertical rod fixed in the field ABCD the angular elevation of top of the tower is âˆ QAP=Î±, âˆ QBP=Î², âˆ QCP=Î³ and âˆ QDP=Î´

Let AL = x and MB = y In right angled Î”ALP, $AP = \sqrt{{AL}^{2} + {LP}^{2}} = \sqrt{x^{2} + y^{2}}$ Similarly, $BP = \sqrt{(l - x)^{2} + y^{2}} CP = \sqrt{(l - x)^{2} + (b - y)^{2}} DP = \sqrt{x^{2} + (b - y)^{2}}$ In right Î”APQ, $\cot α = \frac{AP}{PQ} ∴ \cot^{2} α = \frac{{AP}^{2}}{{PQ}^{2}} = \frac{x^{2} + y^{2}}{{PQ}^{2}}$ In right Î”BPQ, $\cot β = \frac{BP}{PQ} ∴ \cot^{2} β = \frac{{BP}^{2}}{{PQ}^{2}} = \frac{(l - x)^{2} + y^{2}}{{PQ}^{2}}$ In right Î”CPQ, $\cot γ = \frac{CP}{PQ} ∴ \cot^{2} γ = \frac{{CP}^{2}}{{PQ}^{2}} = \frac{(l - x)^{2} + (b - y)^{2}}{{PQ}^{2}}$ In right Î”DPQ, $\cot^{2} δ = \frac{DP}{PQ} \cot^{2} δ = \frac{{DP}^{2}}{{PQ}^{2}} = \frac{x^{2} + (b - y)^{2}}{{PQ}^{2}} \cot^{2} α - \cot^{2} β = \frac{x^{2} + y^{2}}{{PQ}^{2}} - \frac{(l - x)^{2} + y^{2}}{{PQ}^{2}} = \frac{x^{2} - (l - x)^{2}}{{PQ}^{2}} (1) a n d \cot^{2} δ - \cot^{2} γ = \frac{x^{2} + (b - y)^{2}}{{PQ}^{2}} - \frac{{(l - x)}^{2} + (b - y)^{2}}{{PQ}^{2}} = \frac{x^{2} - (l - x)^{2}}{{PQ}^{2}} (2)$ hence from (1) and (2) $\cot^{2} α - \cot^{2} β = \cot^{2} δ - \cot^{2} γ$ hence proved

A vertical rod is fixed in a horizontal rectangular field ABCD.The angular elevations of its top A ,B,C and D areα , β, γ and δ respectively.Show that cot 2 α - cot2 β = cot 2 δ - cot 2 γ

A vertical rod is fixed in a horizontal rectangular field ABCD.The angular elevations of its top A ,B,C and D areα , β, γ and δ

respectively.Show that cot ² α - cot² β = cot ² δ - cot ² γ