Question

express all the trigonometric ratios in terms of tan Θ(with calculations)

Answer 1

the same look at attachement
for cos
cos=1/sectheta=1/(1+tan^2theta)

Answer 2

We are aware of the combinations of trigonometric ratios by expressing one ration into other ratio, such that  

$\sin x=\frac{1}{\cosec x}$

$\cos x=\frac{1}{\sec x}$

$\tan x=\frac{1}{\cot x}$

Trigonometric identities:

$\sin x^{2}+\cos x^{2}=1$

$\sec x^{2}-\tan x^{2}=1$

$\cosec x^{2}-\cot x^{2}=1$

By using the above identities, we can represent one ratio into another format.

$\sin x=\frac{1}{\cosec x}$

$=\sqrt{\frac{1}{\cosec x^{2}}}$

$=\sqrt{\frac{1}{1+\cot x^{2}}}$

$=\sqrt{\frac{\tan x^{2}}{1+\tan x^{2}}}$

$\cos x=\frac{1}{\sec x}$

$=\sqrt{\frac{1}{\sec x^{2}}}$

$=\sqrt{\left(\frac{1}{1+\tan x^{2}}\right)}$  

$\cosec x=\frac{1}{\sin x}$

By using above sine representation the value of $\frac{1}{\sin x}$ is

$=\frac{1}{\sqrt{\left(\frac{\tan x^{2}}{1+\tan x^{2}}\right)}}$

$c\sec x=\sqrt{\sec x^{2}}$

$=\sqrt{1+\tan x^{2}}$

$\cot x=\frac{1}{\tan x}$