(1)

$$\begin{gathered} Since, \\ {csc }^2\theta -{cot }^2\theta =1 \\ \Rightarrow \left(csc\theta -cot\theta \right)\left(csc\theta +cot\theta \right)=1 \\ \Rightarrow csc\theta -cot\theta =\frac{1}{csc\theta +cot\theta } \\ or \\ csc\theta +cot\theta =\frac{1}{csc\theta -cot\theta } \\ \left(csc\theta -cot\theta \right) is reciprocal of\left(csc\theta +cot\theta \right) and vice versa. \end{gathered}$$

So, both Assertion (A) and Reason (R) are true and R is the correct explanation of A.

(3)

Fact given in Assertion (A) i.e. ${sin }^2\theta +{cos }^2\theta =1$

is true because it is a trigonometric identity and satisfies for any value of $\theta$ for which trigonometric ratios involved in it exist.

Hence, assertion (A) is true.

But reason (R) is false as fact given under Reason (R) is not true because a trigonometric identity is satisfied for any value of $\theta$, for which trigonometric ratios involved in it are defined.

(4)

$$\begin{gathered} For assertion \left(A\right), \\ we have, \\ \frac{{cos }^2\theta }{{sin }^2\theta }-\frac{1}{{sin }^2\theta }=\frac{{cos }^2\theta -1}{{sin }^2\theta }=\frac{-\left(1-{cos }^2\theta \right)}{{sin }^2\theta } \\ =\frac{-{sin }^2\theta }{{sin }^2\theta }=-1\left[\because {cos }^2\theta +{sin }^2\theta =1\Rightarrow {sin }^2\theta =1-{cos }^2\theta \right] \end{gathered}$$

Hence, assertion (A) is false.

Also, fact given under reason (R) is well-known identity. So, it is true.

(1)

Both Assertion and Reason are true, and the Reason is the correct explanation of the Assertion. If you divide the basic Pythagorean identity

${sin }^2\theta +{cos }^2\theta =1 by {cos }^2\theta$ then you get $1+{tan }^2\theta ={sec }^2\theta \Rightarrow {sec }^2\theta -{tan }^2\theta =1.$

(3)

Assertion (A) is true, but Reason (R) is false as the identity ${sec }^2\theta -{tan }^2\theta =1$

$is not defined for \theta = 90°$

(3)

Assertion (A) is true, but Reason (R) is false as the identity ${csc }^2\theta -{cot }^2\theta =1 is not defined for \theta = 0°.$

(1)

Hypotenuse is the longest side of the triangle. 
$$\begin{gathered} \therefore sin\theta =\frac{\mathrm{Opposite} \mathrm{side}}{\mathrm{Hypotenuse}}<1\mathrm{and}cos\theta =\frac{\mathrm{Adjacent} \mathrm{side}}{\mathrm{Hypotenuse}}<1 \\ and hence {sin }^2\theta \le 1 \mathrm{and} {cos}^{ }2\theta \le 1. \end{gathered}$$

Thus, both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(1)

Hypotenuse is the longest side of the triangle.

$$\begin{gathered} sin\theta =\frac{\mathrm{Opposite} \mathrm{side}}{\mathrm{Hypotenuse}}<1\mathrm{and}cos\theta =\frac{\mathrm{Adjacent} \mathrm{side}}{\mathrm{Hypotenuse}}<1 \\ and hence {sin }^2\theta \le 1 \&{cos }^2\theta \le 1 \end{gathered}$$

Thus, both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(4)

Assertion (A) is false because if ${sin }^2\theta \le \frac{1}{2},$ then $\sin \theta$ can be equal to 0. In that case, cosθ will be equal to 1$(when \theta =0°).$. However, in the Assertion, the equality sign on cosθ is missing. Hence the Assertion is false.