Question

Prove the following trigonometric identities:
sinA/secA+tanA-1+cotA/cosecA+cotA-1=1

Answer 1

sinA/(secA+tanA-1+cosA)/(cosecA+cotA-1)=1

Step-by-step explanation:

Correction

sinA/(secA+tanA-1+cosA)/(cosecA+cotA-1)=1

LHS

= SinA/(secA+tanA-1)  + cosA/(cosecA+cotA-1)

= SinA/(1/CosA  + SinA/CosA - 1)   + cosA/(1/SinA+cosA/SinA-1)

= SinACosA/(1 + SinA - CosA)  + CosASinA/(1 + CosA - SinA)

= SinACosA/( 1 + (SinA - CosA)  + CosASinA (1 - (SinA - CosA))

= SinACosA(1  - (SinA - CosA)  +  1 + (SinA - CosA) )/( 1² - (SinA - CosA)²)

= SinACosA(2)/(1  - (Sin²A + Cos²A - 2SinACosA)

= 2SinACosA/(1  - (1 - 2SinACosA)

= 2SinACosA/2SinACosA

= 1

= RHS

QED

Proved

sinA/(secA+tanA-1+cosA)/(cosecA+cotA-1)=1

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Answer 2

$\huge\star\mathfrak\blue{{Answer:-}}$

Correction

sinA/(secA+tanA-1+cosA)/(cosecA+cotA-1)=1

LHS

= SinA/(secA+tanA-1) + cosA/(cosecA+cotA-1)

= SinA/(1/CosA + SinA/CosA - 1) + cosA/(1/SinA+cosA/SinA-1)

= SinACosA/(1 + SinA - CosA) + CosASinA/(1 + CosA - SinA)

= SinACosA/( 1 + (SinA - CosA) + CosASinA (1 - (SinA - CosA))

= SinACosA(1 - (SinA - CosA) + 1 + (SinA - CosA) )/( 1² - (SinA - CosA)²)

= SinACosA(2)/(1 - (Sin²A + Cos²A - 2SinACosA)

= 2SinACosA/(1 - (1 - 2SinACosA)

= 2SinACosA/2SinACosA

= 1

= RHS

QED

Proved

sinA/(secA+tanA-1+cosA)/(cosecA+cotA-1)=1