Question

prove that cos theta - sign theta / cos theta + sign theta - 1​

Answer 1

Answer:

→ Hey Mate,

→ Given Question:  prove that   $\frac{costhetha - sin theta +1}{costheta + sintheta - 1}$ = $cosectheta + cottheta$

→ Step-by-step explanation:

→ Solution : - $\frac{costheta - sintheta + 1}{costheta + sintheta -1}$

→              ⇒ Divide Nr and Dr by sin theta

→              ⇒  $\frac{costheta }{sintheta}$   -   $\frac{sintheta}{sintheta}$    + $\frac{1}{sintheta}$

→                 -------------------------------------------     ( Divided By )

→              ⇒ $\frac{costheta}{sintheta}$  +    $\frac{sintheta}{sin theta}$     -  $\frac{1}{sintheta}$

→              ⇒ $cottheta-1 + cosectheta$

→              ⇒ ---------------------------------    ( Divided By )

→              ⇒ $cottheta + 1 - cosectheta$

→              ⇒ [ ∵Formula used: $cosecc^2thetha- cot^2theta = 1$ ]

→              ⇒ $\frac{cosectheta +cottheta - ( cox^2theta - cot^2theta = 1}{(cot theta + 1-cosecthetha) }$

→               ⇒ $\frac{coxcthetha + cot theta - ( coxcthetha + cottheta) ( coxcthetha - cottheta)}{(cottheta+1 - cosxctheta)}$

→               ⇒  $\frac{(coxctheta + cot theta) [ 1 - (coxctheta - cottheta}{ "}$

→               ⇒ $\frac{(coxctheta + cottheta)(1-coxctheta + cottheta )}{(cottheta +1 - coxctheta)}$

→                ⇒ $( coxctheta + cot theta )$ is the solution.

Answer 2

Question :- prove that (cosA-sinA+1) / (cos A+sinA-1) = (cosecA+cotA)

Formula used :-

• CosA/sinA = cotA
• 1/sinA = cosA
• 1 = cosec²A - cot²A
• (a² - b²) = (a+b)(a-b)

Solution :-

→ (cos A- sin A + 1) / (cos A + sin A - 1).

Divide both numerator and the denominator by sinA we get ,,

→ (cosec A + cot A - 1)/(cotA - cosec A +1)

Now, Putting value of 1 = cosec²A - cot²A = (cosecA + cotA)(cosecA - cotA) in Numerator we get,

→ {(cosecA+cotA)-(cosecA+cotA)(cosecA-cotA)} / (cotA - cosec A +1)

Taking (cosecA+cotA) common From Numerator Now, we get,

→ [(cosecA+cotA){1-(cosecA-cotA)}] / (cotA - cosec A +1)

→ (cosecA+cotA)(cotA-cosecA+1) / (cotA - cosec A +1)

(cotA - cosec A +1) will be cancel now

Finally, we get,

→ (cosecA + cotA) { Hence Proved.}

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