Prove the identity:
secθ + cosecθ x cotθ = secθ x (cosecθ)^2

2 thoughts on “Prove the identity:<br /> secθ + cosecθ x cotθ = secθ x (cosecθ)^2”

1. Answer:

   [tex]\huge\colorbox{orange}{ANSWER}[/tex]

   Now,

   (cosecθ−sinθ)(secθ−cosθ)

   =(sinθ1−sinθ)(cosθ1−cosθ)

   =(sinθ1−sin2θ)(cosθ1−cos2θ)

   =sinθcosθcos2θsin2θ=sinθcosθ (1)

   Next, consider tanθ+cotθ1

   =cosθsinθ+sinθcosθ1

   =(sinθcosθsin2θ+cos2θ)1

   =sinθcosθ (2)

   From (1) and (2), we get

   (cosecθ−sinθ)(secθ−cosθ)=tanθ+cotθ

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2. Here I am using A instead of theta.

   We know the trigonometric

   identities ,

   1 ) sin² A + cos² A = 1

   2 ) sec² A = 1 + tan² A

   3 ) cosec² A = 1 + cot² A

   And

   4 ) sinA cosecA = 1

   5 ) cosA secA = 1

   Now ,

   LHS = (sinA+cosecA)² + (cosA+secA)²

   = sin²A+cosec² A+2sinAcosecA+

   cos² A + sec² A + 2cosAsecA

   = (Sin²A + cos² A ) + 2 + 2 + cosec² A

   + Sec² A

   = 1 + 2 + 2 + ( 1 + cot² A )+( 1 + tan²A )

   = 7 + cot² A + tan² A

   = RHS

   I hope this helps you.

   Reply