Question

1) Prove the following identities:

(1 + tan2 A)+(1+1/tan2A) =1/sin2A-sin4A​

Answer 1

Answer:

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Let the L.H.S side

1+tan²A+(1+1/tan²A)

1+Sin²A/Cos²A+ (1+1/Sin²A/Cos²A)

1+Sin²A/Cos²A+(1+Cos²A/Sin²A)

Supposed that,

x=Sin²A

Sin²A=1-Cos²A

x=1-Cos²A

Cos²A=1-x

Therefore,

1+x/(1-x)+{1+(1-x)/x}

{(1-x)+x}/(1-x)+{x+(1-x)}/x

(1-x+x)/(1-x)+(x+1-x)/x

1/(1-x)+1/x

x+(1-x)/x(1-x)

(x+1-x)/x-x²

1/x-x²

Putting x as Sin²A

1/Sin²-Sin⁴A

Hence proved

Answer 2

Answer:

Let the L.H.S side

1+tan²A+(1+1/tan²A)

1+Sin²A/Cos²A+ (1+1/Sin²A/Cos²A)

1+Sin²A/Cos²A+(1+Cos²A/Sin²A)

Supposed that,

x=Sin²A

Sin²A=1-Cos²A

x=1-Cos²A

Cos²A=1-x

Therefore,

1+x/(1-x)+{1+(1-x)/x}

{(1-x)+x}/(1-x)+{x+(1-x)}/x

(1-x+x)/(1-x)+(x+1-x)/x

1/(1-x)+1/x

x+(1-x)/x(1-x)

(x+1-x)/x-x²

1/x-x²

Putting x as Sin²A

1/Sin²-Sin⁴A

Hence , proved !

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