Question

Please ans this prove it​

Answer 1

Refer to the attachment for complete solution.

All Trigonometric Basic Formulas :-

• sin A = Perpendicular / Hypotenuse
• cos A = Base / Hypotenuse
• tan A = Perpendicular / base
• cosec A = Hypotenuse / Perpendicular
• cos A = Hypotenuse / base
• cot A = Base / Hypotenuse
• sec² A - tan² A = 1
• 1 + cot² A = cosec²A
• sin² A + cos² A = 1
• sin A = 1 / cosec A
• cos A = 1 / sec A
• tan A = 1 / cot A
• cosec A = 1 / sin A
• sec A = 1 / cos A
• cot A = 1 / tan A
• tan A = sin A / cos A
• tan A = sec A / cosec A
• cot A = cos A / sin A
• cot A = cosec A / cos A
• sin ( 90° - A ) = cos A
• cos ( 90° - A ) = sin A
• tan ( 90° - A ) = cot A
• cot ( 90° - A ) = tan A
• cosec ( 90° - A ) = sec A
• sec ( 90° - A ) = cos A

Answer 2

$\large\underline{\sf{Given \:Question - }}$

Prove that

$\rm :\longmapsto\:\dfrac{tanA + secA - 1}{tanA - secA + 1} = \dfrac{1 + sinA}{cosA}$

$\large\underline{\sf{Solution-}}$

Consider, LHS

$\rm :\longmapsto\:\dfrac{tanA + secA - 1}{tanA - secA + 1}$

We know that,

$\boxed{ \rm \: {sec}^{2}x - {tan}^{2}x = 1}$

So, using this identity in numerator, we get

$\rm \:  =  \:\dfrac{tanA + secA - ( {sec}^{2} A - {tan}^{2} A)}{tanA - secA + 1}$

We know,

$\boxed{ \rm \: {x}^{2} - {y}^{2} = (x + y)(x - y)}$

So, using this identity, we get

$\rm \:  =  \:\dfrac{(tanA + secA) - (secA + tanA)(secA - tanA)}{tanA - secA + 1}$

$\rm \:  =  \:\dfrac{(secA + tanA)\bigg[1 - (secA - tanA)\bigg]}{tanA - secA + 1 }$

$\rm \:  =  \:\dfrac{(secA + tanA)\bigg[1 - secA + tanA\bigg]}{tanA - secA + 1 }$

$\rm \:  =  \:secA + tanA$

$\rm \:  =  \:\dfrac{1}{cosA} + \dfrac{sinA}{cosA}$

$\rm \:  =  \:\dfrac{1 + sinA}{cosA}$

Hence,

$\rm :\longmapsto\:\boxed{ \bf \:\dfrac{tanA + secA - 1}{tanA - secA + 1} = \dfrac{1 + sinA}{cosA} }$

Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1