Question

(1-cos tita)(1+cos tita)(1+cot²tita)
​

Answer 1

Answer:

$\frac{1+ cos^{2}}{1 - cos^{2} }$                                       (Please do not miss the Ф symbol)                      

Step-by-step explanation:

(1 - cosФ)(1 + cosФ)(1 + cot²Ф)

By using (a-b)(a+b) = (a²-b²),

{(1 - cosФ)(1 + cosФ)}(1 + cot²Ф)

⇒ (1² - cos²Ф) (1 + cot²Ф)

⇒ (1 - cos²Ф) (1 + cot²Ф)      · · · (As 1² = 1)

Now, we can use, (x-a)(x-b) = x² - (a+b)x + ab, · · · (Here x=1, a=cos²Ф, b=cot²Ф)

⇒ 1² - (cos²Ф + cot²Ф)1 + (cos²Ф)(cot²Ф)

⇒ 1 - (cos²Ф + cot²Ф) + (cos²Ф)(cot²Ф)

Wkt, cotФ = cosФ/sinФ, i.e., 1/tanФ

So, let us substitute this in the above equation:

⇒ 1 - (cos²Ф + cot²Ф) + (cos²Ф)(cot²Ф)

⇒ 1 - (cos²Ф + cos²Ф/sin²Ф) + (cos²Ф)(cos²Ф/sin²Ф)

⇒ 1 - (cos²Ф + $\frac{cos^{2}Ф}{sin^{2}Ф }$ ) + (cos²Ф)( $\frac{cos^{2}Ф}{sin^{2}Ф }$ )           (Please do not miss the Ф symbol)

Now, let us use LHS:

⇒ 1 - ($\frac{(cos^{2})(sin^{2})+cos^{2}}{sin^{2} }$) + $\frac{cos^{4} }{sin^{2} }$                           (Please do not miss the Ф symbol)

⇒ 1 - $\frac{sin^{2}cos^{2}+cos^{2}}{sin^{2} }$ +  $\frac{cos^{4} }{sin^{2} }$                                 (Please do not miss the Ф symbol)

The denominators have the same value:

⇒ 1 - $\frac{sin^{2}cos^{2}+cos^{2} + cos^{4}}{sin^{2} }$                                     (Please do not miss the Ф symbol)

Let's bring the "1" into the equation as well:

⇒ $\frac{sin^{2} }{sin^{2} } + \frac{sin^{2}cos^{2}+cos^{2} + cos^{4}}{sin^{2} }$       · · · (The denominator for 1 becomes sin²Ф, so 1 can be represented as sin²Ф/sin²Ф.)

⇒ $\frac{sin^{2} + sin^{2}cos^{2}+cos^{2} + cos^{4}}{sin^{2} }$                                  (Please do not miss the Ф symbol)

We know that sin²Ф + cos²Ф = 1,

⇒ $\frac{(sin^{2} + cos^{2})+ sin^{2}cos^{2}+ cos^{4}}{sin^{2} }$  · · · (I have simple re-arranged the terms.)

⇒ $\frac{(1)+ sin^{2}cos^{2}+ cos^{4}}{sin^{2} }$

⇒ $\frac{1+ sin^{2}cos^{2}+ cos^{4}}{sin^{2} }$                                              (Please do not miss the Ф symbol)

Now, let's take the common term: cos²Ф,

⇒ $\frac{1+ cos^{2}(sin^{2}+ cos^{2})}{sin^{2} }$                                           (Please do not miss the Ф symbol)

Once again, we have sin²Ф + cos²Ф = 1,

⇒ $\frac{1+ cos^{2}(1)}{sin^{2} }$

⇒ $\frac{1+ cos^{2}}{sin^{2} }$                                                           (Please do not miss the Ф symbol)

To simplify the equation further, we can use sin²Ф + cos²Ф = 1, and write

sin²Ф = 1 - cos²Ф, in the denominator: (Re-arranging the terms)

⇒  $\frac{1+ cos^{2}}{1 - cos^{2} }$                                                           (Please do not miss the Ф symbol)

So, the answer is (1 + cos²Ф)/(1 - cos²Ф)

I really hoped you understood it!

All the Best!