Given below are two statement: Statement I: If $\sin (\theta)=\frac{5}{13}$, then the value of $\tan (\theta)=\frac{5}{12}$ Statement II: if $\cot (\theta)=\frac{12}{5}$, then the value of $\sin (\theta)=\frac{5}{12}$ In the light of the above statements, choose the correct answer form the question below:

If $\sin \theta+\cos \theta=\frac{\sqrt{7}}{2}$, then $(\sin \theta-\cos \theta)$ is equal to :

The minimum value of $(2 \sin^2\theta + 3 \cos^2\theta)$ is

Two ships are sailing in the sea on the two sides of a lighthouse. The angles of elevation of the top of the lighthouse as observed from two ships are 30° and 45°, respectively. If the lighthouse is 100 m high, the distance between two ships is approximately:

Given below are two statements:

Statement I: $\frac{\cot 30^{\circ}+1}{\cot 30^{\circ}-1}=2\left(\cos 30^{\circ}+1\right)$

Statement II : $2 \sin 45^{\circ} \cos 45^{\circ}-\tan 45^{\circ} \cot 45^{\circ}=0$

Which of the following trigonometric identities are true?

$\begin{aligned} & \sin ^2\left(41^{\circ}\right)+\sin ^2\left(49^{\circ}\right)=1 \\ & \sin ^2\left(60^{\circ}\right)-2 \tan \left(45^{\circ}\right)-\cos ^2\left(30^{\circ}\right)=-1 \\ & \sin ^2(\theta)+\frac{1}{1+\tan ^2(\theta)}=1 \end{aligned}$

If $\sin (θ) - \cos (θ) = 0$, the value of sin⁴ (θ) + cos⁴ (θ) is: