Trigonometric Functions

$\left(1+\cot \mathrm{\theta }-\mathrm{cosec} \mathrm{\theta }\right)\left(1+\tan \mathrm{\theta }+\sec \mathrm{\theta }\right)$ is equal to_____

If $2 {\cos }^2\theta +\sin \theta -2=0,0<\theta \le \frac{\pi }{2},$ then what is the value of $\theta$?

What is the value of $\left(1+{\cot }^2\theta \right)(1+\cos \theta )(1-\cos \theta )-\left(1+{\tan }^2\theta \right)(1+\sin \theta )(1-\sin \theta )$?

What is the value of $\frac{{\cos }^{2}32°+{\cos }^{2}58°}{{\sec }^{2}50°-{\cot }^{2}40°}+4 \tan 13° \tan 37° \tan 53° \tan 77°$?

What is the value of $x$ that satisfies $4 {\cos }^{2}30°+2x \sin 30°-{\cot }^{2}30°-6 \tan 15° \tan 75°=0$?

What is the minimum value of $6-4\sin \theta ,0\le \theta \le \frac{\pi }{2}$?

If ${\tan }^8\theta +{\cot }^8\theta =m,$ then what is the value of $\tan \theta +\cot \theta$?

If $\sin \theta =\frac{12}{13},$ then what is the value of $(\tan \theta +\sec \theta {)}^2(cosec\theta -\cot \theta {)}^{-2},0<\theta <\frac{\pi }{2}$?

Consider the following:

1. ${\cos }^4\theta -{\sin }^4\theta =\frac{2\tan \theta }{1-{\tan }^2\theta },0<\theta <\frac{\pi }{2}$

2. $cosec\theta +\cot \theta =\frac{1}{cosec\theta -\cot \theta },0<\theta <\frac{\pi }{2}$

3. ${\cos }^2\theta -{\sin }^2\theta =\frac{1-{\tan }^2\theta }{1+{\tan }^2\theta },0<\theta <\frac{\pi }{2}$

Which of the above equations are identities?

If $\mathrm{sin\theta }=\frac{\mathrm{x}}{\mathrm{x}+\mathrm{y}}$, then which one of the following is correct?

If $\tan A+\cot A=5$, then ${\tan }^{4}A+{\cot }^{4}A$ is equal to

Find the value of $\frac{\left(3\sin \left({\displaystyle \frac{3\mathrm{\pi }}{2}-a}\right)-4{\sin }^3\left({\displaystyle \frac{3\mathrm{\pi }}{2}-a}\right)\right)}{\left(3\cos \left(\frac{3\mathrm{\pi }}{2}-a\right)-4{\cos }^3\left(\frac{3\mathrm{\pi }}{2}-a\right)\right)}$ :

What should come in the place of $\left(?\right)$ of the expression given below?

$-\frac{\sin \left(\frac{3\pi }{2}+x\right)}{2\cos \left(\frac{3\pi \hspace{0.17em}}{2}+x\right)}+\frac{\cos \left(\frac{3\pi \hspace{0.17em}}{2}+x\right)}{2\sin \left(\frac{3\pi \hspace{0.17em}\hspace{0.17em}}{2}+x\right)}=?$

What should come in the place of $\left(?\right)$ for the expression given below?

$\frac{{\tan }^3\left({\displaystyle \frac{3\mathrm{\pi }}{2}}+x\right)-3\tan \left({\displaystyle \frac{3\mathrm{\pi }}{2}}+x\right)}{1-3{\tan }^2\left({\displaystyle \frac{3\mathrm{\pi }}{2}}+x\right)}=?$

What should come in the place of $\left(?\right)$ for the expression given below?

$\frac{1+\sin \left({\displaystyle \frac{3\mathrm{\pi }}{2}}+x\right)}{\cos \left({\displaystyle \frac{3\mathrm{\pi }}{2}}+x\right)}-\frac{1-\sin \left({\displaystyle \frac{3\mathrm{\pi }}{2}}+x\right)}{\cos \left({\displaystyle \frac{3\mathrm{\pi }}{2}}+x\right)}-\frac{2}{\tan \left({\displaystyle \frac{3\mathrm{\pi }}{2}}+x\right)}=?$

What should come in the place of $\left(?\right)$ of the expression given below?

$\frac{\left(3\sin \left({\displaystyle \frac{3\mathrm{\pi }}{2}}+x\right)-4{\sin }^3\left({\displaystyle \frac{3\mathrm{\pi }}{2}}+x\right)\right)}{\left(3\cos \left(\frac{3\mathrm{\pi }}{2}+x\right)-4{\cos }^3\left(\frac{3\mathrm{\pi }}{2}+x\right)\right)}=?$

What should come in the place of ($?$) for the expression given below?

$\frac{1-\cos \left(2\mathrm{\pi }-x\right)}{\sin \left(x\right)}-\frac{1-\cos \left(\mathrm{\pi }\hspace{0.17em}+x\right)}{\sin \left(2\mathrm{\pi }+x\right)}+\frac{2}{\tan \left(x\right)}=?$

What should come in the place of ($?$) for the expression given below?

$\left(\frac{4{\left(\cos \left(2\mathrm{\pi }\hspace{0.17em}-\mathrm{\theta }\right)\right)}^3+3\cos \left(\mathrm{\pi }\hspace{0.17em}-\mathrm{\theta }\right)}{\left(3\sin \left(\mathrm{\pi }\hspace{0.17em}-\mathrm{\theta }\right)+4{\sin }^3\left(2\mathrm{\pi }\hspace{0.17em}\hspace{0.17em}-\mathrm{\theta }\right)\right)}\right)\times \left(\frac{3\tan \left(\mathrm{\theta }\right)+{\left(\tan \left(2\mathrm{\pi }-\mathrm{\theta }\right)\right)}^3}{1-3{\tan }^2\left(\mathrm{\pi }-\mathrm{\theta }\right)}\right)=?$

What should come in the place of ($?$) for the expression given below?

$\frac{1+{\tan }^2\left(\mathrm{\theta }\right)}{1+{\cot }^2\left(\mathrm{\theta }\right)}-{\sin }^2\left(\mathrm{\theta }\right){\sec }^2\left(\mathrm{\pi }-\mathrm{\theta }\right)=?$

Find the value of the trigonometric function at the given angles.

$y=\sin \left(\frac{7\mathrm{\pi }}{6}\right)\sec \left(\frac{\mathrm{\pi }}{3}\right)+{\left(\sin \left(\frac{11\mathrm{\pi }}{6}\right)\sec \left(\frac{\mathrm{\pi }}{3}\right)\right)}^7$