Trigonometric Identities

Prove the following identity.

$\sin A(\sec A+\mathrm{cosec}A)-\cos A(\sec A-\mathrm{cosec}A)=\sec A\mathrm{cosec}A$

Prove the following identity.

$\sin A(1+\tan A)+\cos A(1+\cot A)=\sec A+\mathrm{cosec}A$

Prove the following identity.

$\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}=\sin A+\cos A$

Prove the following identity.

$\frac{1+\cos \theta +\sin \theta }{1+\cos \theta -\sin \theta }=\frac{1+\sin \theta }{\cos \theta }$

Prove the following identity.

$\frac{{\sin }^{3}A+{\cos }^{3}A}{\sin A+\cos A}=1-\sin A\cos A$

Prove the following identity.

${\tan }^{2}A+{\cot }^{2}A+2={\sec }^{2}A{\mathrm{cosec}}^{2}A$

Prove the following identity.

$\left(1+\frac{1}{{\tan }^{2}A}\right)\left(1+\frac{1}{{\cot }^{2}A}\right)=\frac{1}{{\sin }^{2}A-{\sin }^{4}A}$

Prove the following identity.

${\sin }^4\theta -{\cos }^4\theta ={\sin }^2\theta -{\cos }^2\theta$

Prove the following identity.

${\cos }^2\theta -{\sin }^2\theta =\frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }$

Prove the following identity.

$\frac{\cos \theta }{\mathrm{cosec}\theta +1}+\frac{\cos \theta }{\mathrm{cosec}\theta -1}=2\tan \theta$

Prove the following identity.

$\frac{1+\cos \theta -{\sin }^2\theta }{\sin \theta (1+\cos \theta )}=\cot \theta$

Prove the following identity.

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

Prove the following identity.

${\sec }^2\theta +{\mathrm{cosec}}^2\theta ={\sec }^2\theta {\mathrm{cosec}}^2\theta$

Prove the following identity.

$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2\sec A$

Prove the following identity.

$\frac{1+\cos A}{\sin A}=\frac{\sin A}{1-\cos A}$

Determine whether the following equation is an identity:

$\tan \Phi \mathrm{cosec}\Phi =\sec \Phi$

Determine whether the following equation is an identity:

${\tan }^2\theta -{\sin }^2\theta ={\tan }^2\theta {\sin }^2\theta$

Determine whether the following equation is an identity:

$\frac{\tan \theta +\sin \theta }{\tan \theta -\sin \theta }=\frac{\sec \theta +1}{\sec \theta -1}$

Determine whether the following equation is an identity:

${\cot }^2\theta +\cos \theta ={\sin }^2\theta$

Prove the following identity.

$\frac{1+\cos \theta }{1-\cos \theta }={(\mathrm{cosec}\theta +\cot \theta )}^2$