Question

tan theta + vot theta = sec theta . cosec theta​

Answer 1

$\red{\huge \: \mathfrak{solution}}$

To prove :

$\tan( \alpha ) + \cot( \alpha ) = \sec( \alpha ) \csc( \alpha )$

LHS

$\implies \: \tan( \alpha ) + \cot( \alpha ) \\ \\ \implies \: \frac{ \sin( \alpha ) }{ \cos( \alpha ) } + \frac{ \cos( \alpha ) }{ \sin( \alpha ) } \\ \\ \leadsto \: \frac{ \sin {}^{2} ( \alpha ) + \cos {}^{2} ( \alpha ) }{ \cos( \alpha ) \sin( \alpha ) } \\ \\ \leadsto \frac{1}{ \cos( \alpha ) \sin( \alpha ) } \\ \\ \leadsto \: \sec( \alpha ) \csc( \alpha )$

LHS = RHS

$\blue{\boxed{proved}}$

Formula used

◆ sin^2@+cos^2@= 1

◆ tan@= sin@/cos@

◆ cot@= cos@/sin@

◆ cos@=1/sec@

◆ sin@= 1/cosec@

Answer 2

Step-by-step explanation:

To Prove : tanθ + cotθ = secθ.cosecθ

L.H.S. : tanθ + cotθ

We know that,

tanθ = sinθ / cosθ

cotθ = cosθ / sinθ

⎇ $\sf{\dfrac{sin\theta }{cos\theta} } + {\dfrac{cos\theta}{sin\theta}}$

⎇ $\sf{\dfrac{ {sin}^{2} \theta \: + \: {cos}^{2} \theta}{cos\theta.sin\theta}}$

Identity : sin²θ + cos²θ = 1

⎇ $\sf{\dfrac{1}{cos\theta.sin\theta}}$

We know that,

1/cosθ = secθ

1/sinθ = cosecθ

⎇ secθ.cosecθ

= R.H.S.

Hence, proved !!