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Find the value of $1 tan A + sin A 1 + cos A$ from the following choices if $cosec A = 2$

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2

12

answer is B.

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Detailed Solution

Given that,

cos e c A = 2

.
Write

cos e c A = 2

in terms of the ratio of sides of the right triangle.

cos e c A = 2 k 1 k = h y p o t e n u s e O p p o s i t e s i d e

We know that, cosec A is the ratio of hypotenuse to the opposite side.
With respect to A, form a triangle with opposite side (BC) as 1k and the hypotenuse (AC) as 2k.
Question Image

Find third side

A B

by applying Pythagoras theorem.

(b a s e) 2 + (p e r p e n d i c u l a r) 2 = (h y p o t e n u s e) 2

⇒ A C 2 = B C 2 + A B 2 ⇒ 4 k 2 = k 2 + A B 2 ⇒ A B 2 = 4 k 2 − k 2 ⇒ A B = 3 k

Write

sin A, tan A

and

cos A

from sides of the triangle for corresponding to angle A.

sin θ = O p p o s i t e s i d e H y p o t e n u s e

⇒ sin A = B C A C ⇒ sin A = 1 k 2 k ⇒ sin A = 12

We know that,

tan θ

is the ratio of opposite side to the adjacent side.

⇒ tan A = B C A B ⇒ tan A = 1 k 3 k ⇒ tan A = 13

And,

cos θ = A d j a c e n t s i d e H y p o t e n u s e

⇒ cos A = A B A C ⇒ cos A = 3 k 2 k ⇒ cos A = 32

Now, find the value of

sin A, tan A and cos A 1 tan A + sin A 1 + cos A

⇒ 113 + 12 1 + 32

Multiply and take LCM.

⇒ 3 + 1 2 + 3 ⇒ 3 (2 + 3) + 1 2 + 3 ⇒ 23 + 3 + 1 2 + 3 ⇒ 23 + 4 2 + 3 ⇒ 2 (2 + 3) 2 + 3 ⇒ 2

Thus,

1 tan A + sin A 1 + cos A = 2

.
Hence, option 2) is correct.

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