Question

Prove that : cot A (1 - cos A) - cosec A ( sec A-1) = sin A-tan A.

plzz guys answer fast​

Answer 1

$\underline{\textsf{To prove:}}$

$\mathsf{cotA(1-cosA)-cosecA(secA-1)=sinA-tanA}$

$\underline{\textsf{Solution:}}$

$\textsf{Consider,}$

$\mathsf{cotA(1-cosA)-cosecA(secA-1)}$

$\mathsf{=\dfrac{cosA}{sinA}(1-cosA)-\dfrac{1}{sinA}(\dfrac{1}{cosA}-1)}$

$\mathsf{=\dfrac{cosA}{sinA}-\dfra{cos^2A}{sinA}-\dfrac{1}{sinA\,cosA}+\dfrac{1}{sinA}}$

$\mathsf{=\dfrac{cosA}{sinA}-\dfra{cos^2A}{sinA}-\dfrac{1}{sinA\,cosA}+\dfrac{1}{sinA}}$

$\mathsf{=\dfrac{1}{sinA}-\dfra{cos^2A}{sinA}+\dfrac{cosA}{sinA}-\dfrac{1}{sinA\,cosA}}$

$\mathsf{=\dfra{1-cos^2A}{sinA}+\dfrac{cos^2A-1}{sinA\,cosA}}$

$\mathsf{=\dfra{1-cos^2A}{sinA}-\dfrac{(1-cos^2A)}{sinA\,cosA}}$

$\mathsf{=\dfra{sin^2A}{sinA}-\dfrac{sin^2A}{sinA\,cosA}}$

$\mathsf{=sinA-\dfrac{sinA}{cosA}}$

$\mathsf{=sinA-tanA}$

$\implies\boxed{\mathsf{cotA(1-cosA)-cosecA(secA-1)=sinA-tanA}}$

Find more:

If sinA+sinB=C,cosA+cosB=D,then value of sin(A+B)=

​https://brainly.in/question/11128542

Answer 2

SOLUTION :

TO PROVE

$\sf{ \cot A(1 - \cos A) - \cosec A( \sec A - 1) }$

$= \sf{ \sin A \: - \tan A}$

PROOF

$\sf{ \cot A(1 - \cos A) - \cosec A( \sec A - 1) }$

$\displaystyle =\sf{ \frac{ \cos A}{ \sin A} \bigg(1 - \cos A \bigg) - \frac{ 1}{ \sin A} \bigg( \frac{1}{ \cos A} - 1 \bigg) }$

$\displaystyle =\sf{ \frac{ \cos A}{ \sin A} \bigg(1 - \cos A \bigg) - \frac{ 1}{ \sin A} \bigg( \frac{1 - \cos A }{ \cos A} \bigg) }$

$\displaystyle =\sf{ \frac{ 1 - \cos A}{ \sin A} \bigg(\cos A - \frac{1}{ \cos A} \bigg) }$

$\displaystyle =\sf{ \frac{ 1 - \cos A}{ \sin A} \bigg( \frac{{\cos}^{2} A - 1}{ \cos A} \bigg) }$

$\displaystyle =\sf{ \frac{ 1 - \cos A}{ \sin A} \bigg( \frac{ - {\sin}^{2} A }{ \cos A} \bigg) }$

$\displaystyle =\sf{ \bigg({ 1 - \cos A}\bigg) \bigg( \frac{ - {\sin} A }{ \cos A} \bigg) }$

$\displaystyle =\sf{ - \frac{ {\sin} A }{ \cos A} + {\sin} A }$

$\displaystyle =\sf{ - {\tan} A + {\sin} A }$

$\displaystyle =\sf{ {\sin} A- {\tan} A }$

Hence proved

━━━━━━━━━━━━━━━━

LEARN MORE FROM BRAINLY

If 3x=cosecθ and 3/x=cotθ,

then find the value of 3(x²-1/x²)

https://brainly.in/question/21134289