Question

Please answer step by step with attachment

Answer 1

Solution :-

→ 1/(cotA + tanA) = sinA * cosA

Taking LHS,

→ 1/(cotA + tanA)

using formula :-

• tanA = (sinA/cosA)
• cotA = (cosA/sinA)

→ 1/{(cosA/sinA) + (sinA/cosA)}

Now Taking LCM of denominators,

→ 1/{(cos²A + sin²A)/(sinA*cosA)}

Now using 1/(a/b) = (b/a) we get,

→ (sinA*cosA)/(sin²A + cos²A)

Now putting sin²A + cos²A = 1 in denominator,

→ (sinA*cosA)/1

→ (sinA*cosA) = RHS .(Proved).

Answer 2

${\huge{\bf{\red{\underline{Solution:}}}}}$

${\bf{\blue{\underline{Given:}}}}$

$\star \: \: {\sf{ \frac{1}{cot \theta + tan \theta} = sin \theta \: cos \theta}}$

${\bf{\blue{\underline{Now:}}}}$

Take L.H.S

$\: \: = \: {\sf{ \frac{1}{cot \theta + tan \theta} }}$

$\dagger \: \: \boxed{\sf{ cot \theta = \frac{cos \theta}{sin \theta} }}$

$\dagger \: \: \boxed{\sf{ tan \theta = \frac{sin \theta}{cos \theta} }}$

put these values in given ,we get

$\:\:= {\sf{ \frac{1}{ \frac{cos \theta}{sin \theta} + \frac{sin \theta}{cos \theta} } }}$

$\:\: = {\sf{ \frac{1}{ \frac{(cos \theta)(cos \theta) + (sin \theta)(cos \theta)}{(sin \theta)(cos \theta)}} }}$

$\: \: = {\sf{ \frac{sin \theta \: cos \theta}{ {sin}^{2} \theta + {cos}^{2} \theta} }}$

$\dagger \boxed {\underline{\sf{ {cos }^{2} \theta + {sin}^{2} \theta = 1}}}$

$\: \: = {\sf{ \frac{sin \theta \: cos \theta}{ 1} }}$

$\: \: = {\sf{ {sin \theta \: cos \theta}}}$

Hence L.HS =R.HS