Question : Which of the following options is NOT a correct trigonometric identity?

Option 1: $1+\cot ^2 \theta =\operatorname{cosec}^2 \theta$

Option 2: $1+\tan ^2 \theta =\sec ^2 \theta$

Option 3: $1-\sin ^2 \theta =\cos ^2 \theta$

Option 4: $\cos ^2 \theta-\sin ^2 \theta=1$

Question : Which of the following is equal to $[\frac{\cos \theta}{\sin \theta}+\frac{\sin \theta}{\cos \theta}]$?

Option 1: $\operatorname{cosec} \theta \sec \theta$

Option 2: $\sec \theta\tan \theta$

Option 3: $\operatorname{cosec} \theta\tan \theta$

Option 4: $\cot \theta \sec \theta$

Question : $\left(\frac{\tan ^3 \theta}{\sec ^2 \theta}+\frac{\cot ^3 \theta}{\operatorname{cosec}^2 \theta}+2 \sin \theta \cos \theta\right) \div\left(1+\operatorname{cosec}^2 \theta+\tan ^2 \theta\right), 0^{\circ}<\theta<90^{\circ}$, is equal to:

Option 1: $\operatorname{cosec} \theta \sec \theta$

Option 2: $\operatorname{cosec} \theta$

Option 3: $\sin \theta \cos \theta$

Option 4: $\sec \theta$

Question : $\frac{(1+\tan \theta+\sec \theta)(1+\cot \theta-\operatorname{cosec} \theta)}{(\sec \theta+\tan \theta)(1-\sin \theta)}$ is equal to:

Option 1: $2 \sec \theta$

Option 2: $2 \operatorname{cosec} \theta$

Option 3: $\operatorname{cosec} \theta$

Option 4: $\sec \theta$

Question : The value of $\sqrt{\frac{1+\cos \theta}{1-\cos \theta}}$ is:

Option 1: $\sec\theta+\tan \theta$

Option 2: $\operatorname{cosec} \theta-\cot \theta$

Option 3: $\operatorname{cosec} \theta+\cot \theta$

Option 4: $\sec\theta-\tan \theta$

Question : If $\tan ^2 \theta+\tan ^4 \theta=1$, then:

Option 1: $\cot ^2 \theta+\cot ^4 \theta=1$

Option 2: $\cos ^2 \theta+\cos ^4 \theta=1$

Option 3: $\sin ^2 \theta+\sin ^4 \theta=1$

Option 4: $\operatorname{cosec}^2 \theta+\sec ^4 \theta=1$