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Find the value of $tan A + tan 2 A − cosec A,$ from the following options, if $tan A = 2$ .

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5 + 8 2

35 + 5 2

35 + 8 2

35 + 8 4

answer is C.

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Detailed Solution

Given that,

tan A = 2

.
Write

tan A = 2

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

tan A = 2

tan A = k 2 k = O p p o s i t e s i d e A d j a c e n t s i d e

With respect to A, form a triangle with opposite side (BC) as 1k and the adjacent side (AB) as 2k.
Question Image

Find third side

A C

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 ⇒ A C 2 = k 2 + 4 k 2 ⇒ A C 2 = 5 k 2 ⇒ A C = 5 k

Write

sec A, tan A

and

cos e c A

from sides of the triangle.

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

⇒ sec A = A C A B ⇒ sec A = 5 k 1 k ⇒ sec A = 5

We know that,

tan θ

is the ratio of opposite side to the adjacent side.

⇒ tan A = B C A B ⇒ tan A = 2 k 1 k ⇒ tan A = 2

We know that,

cos e c A = H y p o t e n u s e O p p o s i t e s i d e

⇒ cos e c A = A C B C ⇒ cos e c A = 5 k 2 k ⇒ cos e c A = 52

To find that value of the expression

sec A . tan A + tan 2 A − cos e c A

⇒ sec A . tan A + tan 2 A − cos e c A = 5 × 2 + 22 − 52

Multiply and take LCM.

⇒ sec A . tan A + tan 2 A − cos e c A = 45 + 8 − 5 2 ⇒ sec A . tan A + tan 2 A − cos e c A = 35 + 8 2

Thus,

sec A . tan A + tan 2 A − cos e c A = 35 + 8 2

.
Hence, option 3) is correct.

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