Question

Prove 1/tanA+cotA=sinAcosA​

Answer 1

Question:-

$\frac{1}{ \tan(A) + \cot(A) } = \sin(A) \cos(A)$

Solution:-

Convert tan(A) and cot(A) in the term of sin(A) and cos(A)

Some trigonometry identities

$\tan(A) = \frac{ \sin(A) }{ \cos(A ) }$

$\cot(A) = \frac{ \cos(A) }{ \sin(A) }$

Now put the value ,

$\to \: \frac{1}{ \tan(A) + \cot(A) }$

$\to \frac{1}{ \frac{ \sin(A) }{ \cos(A) } + \frac{ \cos(A) }{ \sin(A) } }$

Now take a lcm ,we get

$\frac{1}{ \frac{\sin {}^{2} (A) + \cos {}^{2} (A) }{ \sin(A) \times \cos(A) } }$

Use this identity

Sin²(A) + Cos²(A) = 1

$\to \frac{1}{ \frac{1}{ \sin(A) \times \cos(A) } } = \frac{1}{1} \times \sin(A) \cos(A)$

$\to \: \sin(A) \cos(A)$

Hence proved