Question

Prove that Sinθ (1+tanθ)+cosθ (1+cotθ)=secθ+cosecθ .

Answer 1

Solution :

LHS -

sin theta ( 1 + tan theta ) + cos theta ( 1 + cot theta )

> sin theta ( 1 + sin theta/cos theta) + cos theta ( 1 + cos theta/sin theta)

> sin theta ( cos theta + sin theta )/ cos theta + cos theta ( sin theta + cos theta )/sin theta

> [ sin theta/ cos theta ] { cos theta + sin theta } + [ cos theta/sin theta ] { cos theta + sin theta }

> tan theta [ sin theta + cos theta ] + cot theta [ sin theta + cos theta ]

Comparing this with the original expression , there are a lot of similarities :P

Now taking ( sin theta + cos theta ) common

> [ sin theta + cos theta ]{ tan theta + cot theta ]

> [ sin theta + cos theta ][ sin² theta + cos² theta ]/[ sin theta cos theta ]

> [ sin theta + cos theta ][ 1 ]/[ sin theta cos theta ]

> [ sin theta + cos theta ]/[ sin theta cos theta ]

> [ sin theta]/[ sin theta cos theta ] + [ cos theta ]/[ sin theta cos theta ]

> [ 1/cos theta ] + [ 1/sin theta ]

> sec theta + cosec theta .

LHS = RHS

$\orange { \sf{ \underline {\underline { Hence \: Proved \: }}}}$

Answer 2

EXPLANATION.

⇒ Sin∅(1 + tan∅) + Cos∅(1 + cot∅) = sec∅ + cosec∅.

As we know that,

⇒ sin∅ + sin∅ tan∅ + cos∅ + cos∅ cot∅.

Formula of :

⇒ tan∅ = sin∅/cos∅.

⇒ cot∅ = cos∅/sin∅.

⇒ sec∅ = 1/cos∅.

⇒ cosec∅ = 1/sin∅.

Using this formula in equation, we get.

⇒ sin∅ + sin∅(sin∅/cos∅) + cos∅ + cos∅(cos∅/sin∅).

⇒ sin∅ + sin²∅/cos∅ + cos∅ + cos²∅/sin∅.

⇒ cos∅ sin∅ + sin²∅/cos∅ + sin∅ cos∅ + cos²∅/sin∅.

Taking L.C.M in equation, we get.

⇒ sin∅[cos∅ sin∅ + sin²∅] + cos∅[sin∅ cos∅ + cos²∅]/cos∅ sin∅.

⇒ cos∅ sin²∅ + sin³∅ + sin∅ cos²∅ + cos³∅/cos∅ sin∅.

⇒ (sin³∅ + cos³∅) + (cos∅ sin²∅ + sin∅ cos²∅)/cos∅ sin∅.

⇒ (sin³∅ + cos³∅) + sin∅ cos∅(sin∅ + cos∅)/cos∅ sin∅.

As we know that,

Formula of :

⇒ (x³ + y³) = (x + y)(x² - xy + y²).

Using this formula in equation, we get.

⇒ (sin∅ + cos∅)(sin²∅ + cos²∅ - sin∅ cos∅) + sin∅ cos∅(sin∅ + cos∅)/cos∅ sin∅.

⇒ (sin∅ + cos∅)(sin²∅ + cos²∅ - sin∅ cos∅ + sin∅ cos∅)/cos∅ sin∅.

⇒ (sin∅ + cos∅)/cos∅ sin∅.

⇒ sin∅/cos∅ sin∅ + cos∅/cos∅ sin∅.

⇒ 1/cos∅ + 1/sin∅.

⇒ sec∅ + cosec∅.

Hence proved.

                                                                                                                       

MORE INFORMATION.

(1) = sin²∅ + cos²∅ = 1.

(2) = 1 + tan²∅ = sec²∅.

(3) = 1 + cot²∅ = cosec²∅.