Can someone please help me with this trigonometry problem!

1.	Can someone please help me with this trigonometry problem!
Answer» 2 Step-by-step explanation: Given: cosec A = 2 To find: The value of $\dfrac{1}{ \tan A} + \dfrac{ \sin A}{1 + \cos A}$ Solution: cosec A = 2 As, We KNOW cosec 30° = 2 So, cosec A = cosec 30° A = 30° Now, $\frac{1}{ \tan A} + \frac{ \sin A}{1 + \cos A} \\ \\ \frac{1}{ \tan30} + \frac{ \sin30}{1 + \cos 30} \\ \\ \cot30 + \frac{ \frac{1}{2} }{1 + \frac{ \sqrt{3} }{2} } \\ \\ \sqrt{3} + \frac{ \frac{1}{2} }{ \frac{2 + \sqrt{3} }{2} } \\ \\ \sqrt{3} + \frac{1}{2} \times \frac{2}{2 + \sqrt{3} } \\ \\ \sqrt{3} + \frac{1}{2 + \sqrt{3} } \\ \\ [\text{Rationalizing the denominator}] \\ \\ \sqrt{3} \: + \frac{(2 - \sqrt{3}) }{(2 + \sqrt{3} )(2 - \sqrt{3} )} \\ \\ \sqrt{3} + \frac{2 - \sqrt{3} }{(2){}^{2} - {( \sqrt{3} )}^{2} } \\ \\ \sqrt{3} + \frac{2 - \sqrt{3} }{4 - 3} \\ \\ \sqrt{3} + 2 - \sqrt{3} \\ \\ 2$ Hence, The required answer is 2. Some IMPORTANT trigonometry RULES: sinø . cosecø = 1 cosø . secø = 1 tanø . cotø = 1 sin²ø + cos²ø = 1 cosec²ø - cot² = 1 sec²ø - tan²ø = 1 Trigonometry Value See in the ATTACHMENT.

Answer»

Step-by-step explanation:

Given:

cosec A = 2

To find:

The value of

$\dfrac{1}{ \tan A} + \dfrac{ \sin A}{1 + \cos A}$

Solution:

cosec A = 2

As, We KNOW

cosec 30° = 2

So, cosec A = cosec 30°

A = 30°

Now,

$\frac{1}{ \tan A} + \frac{ \sin A}{1 + \cos A} \\ \\ \frac{1}{ \tan30} + \frac{ \sin30}{1 + \cos 30} \\ \\ \cot30 + \frac{ \frac{1}{2} }{1 + \frac{ \sqrt{3} }{2} } \\ \\ \sqrt{3} + \frac{ \frac{1}{2} }{ \frac{2 + \sqrt{3} }{2} } \\ \\ \sqrt{3} + \frac{1}{2} \times \frac{2}{2 + \sqrt{3} } \\ \\ \sqrt{3} + \frac{1}{2 + \sqrt{3} } \\ \\ [\text{Rationalizing the denominator}] \\ \\ \sqrt{3} \: + \frac{(2 - \sqrt{3}) }{(2 + \sqrt{3} )(2 - \sqrt{3} )} \\ \\ \sqrt{3} + \frac{2 - \sqrt{3} }{(2){}^{2} - {( \sqrt{3} )}^{2} } \\ \\ \sqrt{3} + \frac{2 - \sqrt{3} }{4 - 3} \\ \\ \sqrt{3} + 2 - \sqrt{3} \\ \\ 2$

Hence, The required answer is 2.

Some IMPORTANT trigonometry RULES:

sinø . cosecø = 1

cosø . secø = 1

tanø . cotø = 1

sin²ø + cos²ø = 1

cosec²ø - cot² = 1

sec²ø - tan²ø = 1

Trigonometry Value

See in the ATTACHMENT.