Step-by-step explanation:

Given:-

In a right angled triangle ABC, right angled at B, given 15 cos A - 8 Sin A = 0.

To find:-

Find the following:

1. Sin A + Cos A / 2 Cos A - Sin A = ?

2. 15 cot A + 17 sin A / 2 Cos A - Sin A = ?

Solution:-

Given that

15 cos A - 8 Sin A = 0

=> 15 Cos A = 8 Sin A

=> Cos A / Sin A = 8/15

=> Cot A = 8/15-------(1)

On SQUARING both sides

=> Cot^2 A = (8/15)^2

=> Cot^2 A = 64/225

Now

=> 1+Cot^2 A = 1+(64/225)

=> 1+ Cot^2 A = (225+64)/225

=> 1+Cot^2 A = 289/225

We know that

Cosec^2 A - Cot^2 A = 1

=> Cosec^2 A = 1+Cot^2 A

=> Cosec^2 A = 289/225

=> Cosec A = √(289/225)

=> Cosec A = 17/15

Sin A = 15/17---------(2)

On squaring both sides

=> Sin^2 A = 225/289

=> 1- Sin^2 A = 1-(225/289)

=> 1-Sin^2 A = (289-225)/289

=> 1- Sin^2 A = 64/289

We know that

Sin^2 A + Cos^2 A = 1

=> Cos^2 A = 64/289

=> Cos A =√(64/289)

Cos A = 8/17-----------(3)

I) The value of

(Sin A + Cos A) / (2 Cos A - Sin A )

=>[ (15/17)+(8/17)]/[2(8/17)-(15/17)]

=>[ (15+8)/17 ]/ [ (16-15)/17]

=> (23/17)/(1/17)

=> 23/1

=>23

(Sin A + Cos A) / (2 Cos A - Sin A ) = 23

2) The value of

(15 cot A + 17 sin A) / (2 Cos A - Sin A)

=> [15(8/15)+17(15/17)] /[ 2(8/17)-(15/17)]

=> [(15×8/15)+(17×15/17)] / [(2×8/17)-(15/17)]

=> (8+15)/ [(16/17)-(15/17)]

=> 23/[(16-15)/17]

=> 23/(1/17)

=> 23×(17/1)

=> 23×17

=> 391

(15 cot A + 17 sin A) / (2 Cos A - Sin A) = 391

Answer:-

The values of

i)(Sin A + Cos A) / (2 Cos A - Sin A ) = 23

ii)(15 cot A + 17 sin A) / (2 Cos A - Sin A) = 391

Used formulae:-

• Cosec^2 A - Cot^2 A = 1

• Sin^2 A + Cos^2 A = 1

• Cot A = Cos A / Sin A

• Cosec A = 1/ Sin A