Assertion (A): The tangent of {45}^\circ\ is 1. Reason (R): The cosine of {90}^\circ\ is 0.

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Solution

Q.
$A s s e r t i o n (A) : T h e t a n g e n t o f 45 ° i s 1 .$

$R e a s o n (R) : T h e c o s i n e o f 90 ° i s 0 .$

1 Both A and R are true, and R is the correct explanation of A

2 Both A and R are true, but R is not the correct explanation of A

3 A is true, but R is false

4 A is false, but R is true

Ans.
(1)

$I n △ A B C, r i g h t a n g l e d a t B, l e t ∠ C = 45 ° . \Rightarrow ∠ A = 90 ° - 45 ° = 45 °$

⇒ ∠A = ∠C

⇒ AB = BC = x (say)

[? In a triangle, sides opposite to equal angles are equal]

$t a n C = t a n 45^{\circ} = A B / B C \dots (i)$

[where AB is perpendicular (opposite side) and BC is base (adjacent side)]

$t a n 45^{\circ} = x / x = 1 [∵ A B = B C = x]$

Hence, assertion A is true.

Also reason (R) is true and represents a fact (that for an angle of 45° in a right triangle, the opposite side equals the adjacent side), supporting assertion (A), so it correctly explains the assertion (A).

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