(1)       Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A)

(4)

$cosec \theta$ is defined as ratio of hypotenuse and opposite side (perpendicular). As hypotenuse is the largest side in any right-angled triangle, the value of $cosec \theta$ cannot be less than 1. However, reason is true

(1)

Both the assertion and the reason are correct. The secant of an angle is indeed the reciprocal of the cosine, and the definition provided in the reason accurately reflects this relationship.

(1)

We know, 

$$\begin{gathered} cot\theta =\frac{1}{\tan \theta } \\ \therefore tan\theta ·cot\theta =tan\theta ·\frac{1}{\tan \theta }=1 \\ So, Reason is correct explanation of Assertion \end{gathered}$$

(1)

$$\begin{gathered} In △ABC right angled at B, let \angle C = \theta \therefore \\ \cos \theta =BC/AC=b/h \end{gathered}$$

[where b stands for base (adjacent side) and h stands for hypotenuse]

Obviously, by keeping hypotenuse constant

As $\theta$ increases ⇒ b decreases

$$\begin{gathered} \Rightarrow b/h decreases \\ \Rightarrow \cos \theta decreases. \end{gathered}$$

Hence, Assertion (A) is true. Also Reason (R) represents the same facts and supports the Assertion (A). Therefore, Reason (R) is true and correctly explains the Assertion (A).

(1)

The assertion is correct as the sine of $90°$is indeed 1.
The reason is also true and correctly explains why this is the case because at $90°$, the right triangle’s opposite side becomes the hypotenuse.

(1)

$$\begin{gathered} In △ABC, right angled at B, let \angle C = 45°. \\ \Rightarrow \angle A = 90° - 45° = 45° \end{gathered}$$

⇒ ∠A = ∠C

⇒ AB = BC = x (say)

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

$tan C=tan {45}^{\circ }=AB/BC\dots \left(i\right)$

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

$tan {45}^{\circ }=x/x=1[\because AB=BC=x]$

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).

(2)

Both statements are true, but the reason does not correctly explain the assertion. The assertion’s truth relies on the fact that the maximum value of sum of sine of A and B occur when A and B are each $45°$

(4)

The assertion is false because the sine of angles that add up to $90°$ are equal to its cosine and vice-versa. The reason is true, as it describes the fundamental property of angle sums in a right triangle.

(1)

$$\begin{gathered} In \Delta ABC, right angled at B, let \angle C = 45°. \\ \Rightarrow \angle A = 90° – 45° = 45° \Rightarrow \angle A = \angle C \\ \Rightarrow AB = BC = x \left(say\right) \\ [\because In a triangle, sides opposite to equal angles are equal] \\ \therefore \tan C = \tan 45° = AB / BC \dots \left(i\right) \\ \hspace{1em}[where AB is perpendicular (opposite side) and BC is base (adjacent side\left)\right] \\ \Rightarrow \tan 45° = x / x = 1 \\ \hspace{1em}[\because AB = BC = x] \end{gathered}$$

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).