Question

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If secA+tanA = 2 then show that cosecA + cotA = 3.​

Answer 1

Answer:

Step-by-step explanation:

Given :

sec A + tan A = 2,   ...(i)

Then, show that cosec A + cot A = 3

Solution :

We know that,

sec² A - tan² A = 1

So,

(sec A + tan A)(sec A - tan A) = 1

(2) (sec A - tan A) = 1

⇒ sec A - tan A = $\frac{1}{2}$ ...(ii)

Adding 1 & 2,

⇒ (sec A + tan A) + (sec A - tan A) = 2 + $\frac{1}{2}$

⇒ 2 sec A = $\frac{5}{2}$

⇒ sec A = $\frac{5}{4}$

⇒ sec A = $\frac{1}{cos A}$

⇒ cos A = $\frac{1}{(\frac{5}{4})}=\frac{4}{5}$

We know that,

sin² A + cos² A = 1

⇒ sin² A = 1 - cos² A

⇒ sin² A = 1 - $(\frac{4}{5})^2$

⇒ sin² A = 1 - $\frac{16}{25}$

⇒ sin² A = $\frac{25-16}{25}$

⇒ sin² A = $\frac{9}{25}=(\frac{3}{5})^2$

⇒ sin A = $\frac{3}{5}$

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cosec A + cot A = $\frac{1}{sin \ A} + \frac{cos \ A}{sin \ A}=\frac{1}{(\frac{3}{5})} + \frac{(\frac{4}{5})}{(\frac{3}{5})} = \frac{5}{3}+\frac{4}{3}=\frac{9}{3}=3$

Hence, proved.

Answer 2

Given :

sec A + tan A = 2,   ...(i)

Then, show that cosec A + cot A = 3

Solution :

We know that,

sec² A - tan² A = 1

So,

(sec A + tan A)(sec A - tan A) = 1

(2) (sec A - tan A) = 1

⇒ sec A - tan A = ...(ii)

Adding 1 & 2,

⇒ (sec A + tan A) + (sec A - tan A) = 2 +

⇒ 2 sec A =

⇒ sec A =

⇒ sec A =

⇒ cos A =

We know that,

sin² A + cos² A = 1

⇒ sin² A = 1 - cos² A

⇒ sin² A = 1 -

⇒ sin² A = 1 -

⇒ sin² A =

⇒ sin² A =

⇒ sin A =

__

cosec A + cot A =

Hence, proved.