Question

Prove each of these identities.

Answer 1

Answer:

➲ ( 1 + sec x )( cosec x - cot x ) = tan x

• Solving for L.H.S

$\implies\quad \sf{(1+sec\:x)(cosec\:x-cot\:x) }$

$\implies\quad \sf{ \left(1+\dfrac{1}{cos\:x}\right)\left(\dfrac{1}{sin\:x}-\dfrac{cos\:x}{sin\:x}\right)}$

$\implies\quad \sf{ \left(\dfrac{1+cos\:x}{cos\:x}\right)\left(\dfrac{1-cos\:x}{sin\:x}\right)}$

$\implies\quad \sf{ \left(\dfrac{1-cos^2 x}{cos\:x.sin\:x}\right)}$

$\implies\quad \sf{ \left( \dfrac{sin^2 x}{cos\:x.sin\:x}\right)}$

$\implies\quad \sf{ \left( \dfrac{sin\:x.\cancel{sin\:x}}{cos\:x.\cancel{sin\:x}}\right)}$

$\implies\quad \sf{\left( \dfrac{sin\:x}{cos\:x}\right) }$

$\implies\quad\underline{\underline{\pmb{ \sf{tan\:x}}} }$

➲ ( 1+ sec x )( 1- cos x ) = sin x. tan x

• Solving for L.H.S

$\implies\quad \sf{ ( 1+ sec\:x)(1-cos\:x)}$

$\implies\quad \sf{\left(1+\dfrac{1}{cos\:x} \right) \left( 1-cos\:x\right)}$

$\implies\quad \sf{ \left(\dfrac{cos\:x+1}{cos\:x} \right)(1-cos \:x)}$

$\implies\quad \sf{\dfrac{1-cos^2 x}{cos\:x} }$

$\implies\quad \sf{\dfrac{sin^2x}{cos\:x} }$

$\implies\quad \sf{sin\:x.\left( \dfrac{sin\:}{cos\:x}\right) }$

$\implies\quad\underline{\underline{\pmb{ \sf{sin\:x.tan\:x}}} }$