Question

Prove each of these identities.

a (1 + secx) (cosec x - cotx) = tan x
b (1 + sec x)(1 - cos x) = sin x tan x ​

Answer 1

➲ ( 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+(1)/(cos\:x)\right)\left((1)/(sin\:x)-(cos\:x)/(sin\:x)\right)}`

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

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

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

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

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

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

`\implies\quad \sf{(1-cos^2 x)/(cos\:x) }`

`\implies\quad \sf{(sin^2x)/(cos\:x) }`

`\implies\quad \sf{sin\:x.\left( (sin\:)/(cos\:x)\right) }`

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