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If $cot θ = 78,$ then what is the value of $(1 + sin θ) (1 − sin θ) (1 + cos θ) (1 − cos θ) ?$

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4964

4862

2964

None of the above.

answer is A.

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Detailed Solution

Given that

cot θ = 78

.
Cotangent is a ratio of the side adjacent to the angle

θ

by side opposite to the angle

θ

cot θ = 78 ⇒ Adjacent side Opposite side = 7 k 8 k

where k is the positive integer.
Form a triangle with respect to

θ

with adjacent side as AB and hypotenuse AC.
Question Image

By applying Pythagoras theorem,

B C 2 = A C 2 + A B 2 ⇒ B C 2 = 64 k 2 + 49 k 2 ⇒ B C 2 = 113 k 2 ⇒ B C = 113 k

cos θ = Adjacent side H y p o t e n u s e ⇒ cos θ = A B B C ⇒ cos θ = 7 k 113 k ⇒ cos θ = 7113

sin θ = Opposite side H y p o t e n u s e ⇒ sin θ = A C B C ⇒ sin θ = 8 k 113 k ⇒ sin θ = 8113

Then,

(1 + sin θ) (1 − sin θ) (1 + cos θ) (1 − cos θ)

is given by,

⇒ 1 + 8113 1 − 8113 1 + 7113 1 − 7113

(a − b) (a + b) = a 2 − b 2

⇒ 1 − 64113 1 − 49113

⇒ 113 − 64 113 × 113 113 − 49 ⇒ 4964

Thus,

(1 + sin θ) (1 − sin θ) (1 + cos θ) (1 − cos θ) = 4964

.
Therefore, option 1 is correct.

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