Prove the Laws of Sines and Cosines and use them to solve problems

If $2\cos \left(\theta \right)-\sin \left(\theta \right)=\frac{1}{\sqrt{2}}$ and $\left(0°<\theta <90°\right)$, then the value of $2\sin \left(\theta \right)+\cos \left(\theta \right)$ is:

If $\cos x+{\cos }^{2}x=1$, then what is the value of ${\sin }^{4}x+{\sin }^{2}x$?

What is the value of $\left(\sin 43° \cos 47°+\cos 43° \sin 47°\right)$?

If $\tan \left(\frac{\mathrm{\pi }}{2}-\frac{\theta }{2}\right)=\sqrt{3}$, then value of $\cos \left(\theta \right)$ is:

If $\theta$ is a positive acute angle and $\tan \left(2\theta \right)\tan \left(3\theta \right)=1$, then the value of the expression $\left(2{cos}^2\frac{5\theta }{2}-1\right)$.

If $\mathrm{sin\theta }+\mathrm{cosec\theta }=2$, then what is the value of ${\sin }^2\mathrm{\theta }+{\mathrm{cosec}}^5\mathrm{\theta }$ when $0°\le \theta \le 90°$?

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

$\cot \left(10°\right)·\cot \left(20°\right)·\cot \left(60°\right)·\cot \left(70°\right)·\cot \left(80°\right)=?$

If ${\tan }^2\left(\theta \right) =1 – a^2$, then find the value of $\sec \left(\theta \right) + {\tan }^3\left(\theta \right)\mathrm{cosec}\left(\theta \right)$.

If $\sec \left(\theta \right)+\tan \left(\theta \right)=5$, then the value of $\frac{\tan \left(\theta \right)+1}{\tan \left(\theta \right)-1}$ is:

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

${\sin }^{2}5°+{\sin }^{2}10°+{\sin }^{2}15°+.......+{\sin }^{2}85°+{\sin }^{2}90°=?$

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

$\frac{\sin \left(\theta \right)-\cos \left(\theta \right)+1 }{\sin \left(\theta \right)+\cos \left(\theta \right)-1} =?$

If $\frac{\cos \alpha }{\cos \beta }=\mathrm{m}$ and $\frac{\cos \alpha }{\sin \beta }=\mathrm{n}$, then value of $({\mathrm{m}}^2+{\mathrm{n}}^2) {\cos }^2\beta$_____.

If $\sin \left(\theta +18°\right)=\cos 60°$ for $\left(0°<\theta <90°\right)$, then find the value of $\cos \left(5\theta \right)$.

Find the value $3\sin \left(20°\right)-4{\sin }^3\left(20°\right)$:

If $\tan \left(2\theta \right)· \tan \left(3\theta \right) =1$, where $0°< \theta < 90°$, then find the value of $\theta$.

If $0º< \theta < 90º$, the value of $(\sin \theta + \cos \theta )$ is _____.

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

$\sin 20°\times \sin 40°\times \sin 60°\times \sin 80° = ?$

If ${\tan }^2\left(\theta \right)=1-e^2$, then the value of $\sec \left(\theta \right)+{\tan }^3\left(\theta \right)\csc \left(\theta \right)$ is

Prove that : ${\sec }^2\left({\tan }^{-1} 2\right)+{\mathrm{cosec}}^2\left({\cot }^{-1} 3\right)=15$

Prove that: ${\cos }^{-1 }\frac{1}{\sqrt{5}}+{\cos }^{-1 }\frac{{\displaystyle 2}}{{\displaystyle \sqrt{5}}}=\frac{\mathrm{\pi }}{2}$