**Example 1:**Solve the equation given as cotx/cosecx + tan x?

For solving this type of equation, we need to convert the trigonometric variable into simplest form. = cotx/cosecx,

We can write cotx as 1/tanx,

= cosecx as 1/sinx, and

= tanx as sinx/cosx,

Put this value in the above equation and we will get:

= (1/tanx )/ (1/sinx) +sinx/cosx,

Now, tanx can be written as sinx/cosx ,on putting this value in the above equation as:

= 1/(sinx/cosx) / (1/sinx) +sinx/cosx

We can rewrite the equation as:

= cosx/sinx *sinx + sinx/cosx,

Now, sinx gets cancel with sinx and we will get the required answer as cosx.+sinx/cosx.

The required simplified equation is (cosx +sinx)/cosx.

In this way, we can simplify any expression.

**Example 2:**Solve the equation given as: (cotx +tanx ) *cosx ?

**Solution: =**(cotx +tanx ) *cosx,

= cotx can be written as 1/tanx, so we can rewrite the given equation as:

= (1/tanx +tanx)*cosx,

Now, tanx can be written as sinx /cosx,

= (1/(sinx/cosx) + (sinx /cosx))*cosx,

= (cosx/sinx +sinx/cosx )*cosx,

Now, on taking LCM we will get:

= (cos

^{2}x+sin

^{2}x/cosx*sinx)*cosx,

Now, we can write sin

^{2}x +cos

^{2}x=1

So, we can write above equation as:

= (1/cosx*sinx) *cosx,

= (1/cosx *1/sinx)*cosx,

= (secx *cosecx)*cosx,

Now, we can write cosx as 1/secx so, we will rewrite the equation as:

= secx *cosecx *1/secx,

So, secx gets cancel with secx and we will get the solution as:

= cosecx.

**Solve the equation given as: cotx + 1 / cosecx ?**

Example:

Example:

**Solution:**cotx can be written as 1/tanx,

tanx can be written as sinx/cosx,

cotx can be written as cosx/sinx, and

We can write cosecx as 1/sinx.

Now, putting all these values in the given equation we will get:

= cosx /sinx +1/1/sinx,

We can write this equation as:

= (cosx +sinx ) * sinnx *1/sinx,

So, sinx gets cancel with sinx and we will get:

= cosx + sinx,

So, the required equation can be simplified as cosx + sinx.