Question

Prove the following:

cosec A
___________ = cos A
cot A + tan A

if u know then only answer,,,
I'll mark correct answer as brainliest ​

Answer 1

Answer:

LHS -

$\frac{ \csc \: a }{ \cot \: a + \tan \: a } \\ = \frac{ \frac{1}{ \sin \: a } }{ \frac{ \cos \: a }{ \sin \: a } + \frac{ \sin \: a }{ \cos \: a } } \\ = \frac{1}{ \sin \: a ( \frac{( { \cos \: a) }^{2} + ( { \sin \: a) }^{2} }{ \sin \: a \: \cos \: a } )} \\ = \frac{1}{ \frac{1}{ \cos \: a \: } }$

[ As, (cos a)^2 + (sin a)^2 = 1 ]

= cos a

Hence proved

Answer 2

STEP-BY-STEP EXPLANATION:

Given LHS is cosecA/(cotA + tanA) and RHS is cosA. We need to prove that LHS is equal to RHS.

(cosecA)/(cotA + tanA) = cosA

Taking LHS:

→ cosecA/(cotA + tanA)

Using trigonometry formulas:

• cosecA = 1/sinA
• secA = 1/cosA
• cotA = 1/tanA = cosA/sinA
• tanA = sinA/cosA
• sinA = 1/cosecA
• cosA = 1/secA
• sin²A + cos²A = 1
• tan²A + 1 = sec²A
• 1 + cot²A = cosec²A

Substitute the values using above formulas,

→ (1/sinA)/(cosA/sinA + sinA/cosA)

→(1/sinA)/[(cos²A + sin²A)/(sinA cosA)]

→ (1/sinA)/[1/(sinA cosA)]

→ 1/sinA × sinA cosA

→ (sinA cosA)/sinA

→ cosA

From above LHS is cosA and RHS is also cosA. Therefore, LHS is equal to RHS.

Hence, proved.