Question

if tan theta +4=3(4cot+1),where 0<theta<90,then the value of 15/sec theta cosec theta is
a)4
b)6
c)5.2
d)4.5​

Answer 1

Answer:

Option d

Step-by-step explanation:

Given :-

Tan θ+4=3(4cot θ+1),where 0°<θ<90°

To find :-

Find the value of 15/(Sec θ Cosec θ) ?

Solution :-

Given that

Tan θ+4=3(4cot θ+1)

=> Tan θ+4 = 12 cot θ + 3

=> Tan θ - 12 cot θ = 3-4

=> Tan θ - 12 cot θ = -1

=> Tan θ +1 = 12 cot θ

=> (Tan θ+1)/Cot θ = 12

=> (Tan θ+1)/(1/Tan θ) = 12

=> (Tan θ+1)(Tan θ) = 12

=> Tan² θ + Tan θ = 12

=> Tan² θ + Tan θ - 12 = 0

=> Tan² θ + 4 Tan θ -3 Tan θ -12 = 0

=> Tan θ ( Tan θ+4) -3( Tan θ +4) = 0

=> (Tan θ+4)(Tan θ-3) = 0

=> Tan θ+4 = 0 or Tan θ -3 = 0

=> Tan θ = -4 or Tan θ = 3

Given that 0°<θ<90°

So Tan θ = 3 ---------(1)

Now

On squaring both sides then

=> Tan² θ = 3²

=> Tan² θ = 9

On adding 1 both sides then

=> 1+Tan² θ = 9+1

=>Sec² θ = 10

Since Sec² θ - Tan² θ = 1

=> Sec θ = √10

and

Tan θ = 3

=> Cot θ = 1/3

On squaring both sides

=> Cot² θ = (1/3)²

=> Cot² θ = 1/9

On adding 1 both sides then

=> 1+ Cot² θ = 1+(1/9)

=> Cosec² θ = (9+1)/9

Since , Cosec² θ - Cot² θ = 1

=> Cosec² θ = 10/9

=> Cosec θ = √(10/9)

=> Cosec θ = √10/3

Now

The value of 15/(Sec θ Cosec θ)

=> 15/[(√10)(√10/3)]

=> 15/(10/3)

=> 15×(3/10)

=> (15×3)/10

=> 45/10

=> 9/2

=> 4.5

Answer :-

The value of 15/(Sec θ Cosec θ) is 4.5

Used formulae:-

→ Sec² θ - Tan² θ = 1

→ Cosec² θ - Cot² θ = 1

→ Cot θ = 1/Tan θ

Used Method :-

→ Splitting the middle term

Answer 2

Step-by-step explanation:

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