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In the given figure, $BC = 15 cm$ and $sin B = 45$ , then from the following options, the value of $tan 2 B − 1 cos 2 B$ is:

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-1

2

answer is C.

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Detailed Solution

Given that,

B C = 15 c m

and

sin B = 45

Write

sin B = 45

in terms of the ratio of sides.

sin B = 45

⇒ A C A B = 4 k 5 k

We know that,

sin θ

is the ratio of opposite side (AC) to the hypotenuse(AB).
Apply Pythagoras theorem to find the value of

k

.
In a right triangle

A B C

(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 B 2 = B C 2 + A C 2 ⇒ 25 k 2 = 225 + 16 k 2 ⇒ 25 k 2 − 16 k 2 = 225 ⇒ 9 k 2 = 225 ⇒ k 2 = 2259 ⇒ k 2 = 25 ⇒ k = ± 5

Discard the negative value of

k

.
Thus,

k = 5

.
So,

A B = 5 k = 5 (5) = 25 c m

A C = 4 k = 4 (5) = 20 c m

Find the value of

tan B, cos B

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 B = B C A B

⇒ cos B = 1525 ⇒ cos B = 35

And,

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

⇒ tan B = A C B C ⇒ tan B = 2015 ⇒ tan B = 43

Now, find the value of

tan 2 B − 1 cos 2 B

The value of

tan 2 B − 1 cos 2 B

is -1.
Hence, option 3) is correct.

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