If A is in the fourth quadrant and cos A = 5/13 find the value of

13 sin A+5 sec A divided by 5 tan A + 6cosec A

As in fourth quadrant cos is positive .
So cosA = 513
So sinA  = (1- cos2A)1/21213
But in fourth quadrant sin is negative , so we will take -12/13
And secA  = 13/5
tanA = sinA/cosA = -12/5
cosecA = 1/sinA = -13/12

Hence 13sinA +5secA5tanA + 6coescA =13×-1213 +5×1355×-125 +6×-1312 =-237

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Since A is in fourth quadrant sin A , cosec A tan A are negative while while sec A is positive.

Now cos A = 5/13 = sec A = 1/cos A = 1/(5/13)

= sec A = 13/5

sec²A = 1 + tan²A = tan²A = sec²A - 1

= tan²A = (13/5)² - 1 ( Putting value of sec A)

= tan²A = (169/25) - 1 = tan²A = (169 - 25)/25

= tan²A = 144/25 = tan A = √(144/25)

= tan A = - 12/5 (As tan A is negative)

Also sin A = tan A x cos A

sin A =( -12/5) x (5/13)

= sin A = -12/13 ( As sin A is negative)

cosec A = 1/sin A = cosec A = 1/(-12/13)

cosec A = -13/12

Now putting these values in (13 sin A + 5 sec A) / (5 tan A + 6 cosec A) we get

{ 13(-12/13) + 5(13/5) } / { 5(-12/5) + 6(-13/12)}

= (- 12 + 13) / ( - 12 - 13/2)

= 1/{ (-24 - 13)/2 }

= 1/ (-37/2)

= 2/- 37

= -2 /37

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