Question

OR
Prove that (Sin A+ CosecA)? +(CosA+Sec A) = 7+ Tan’A+Cot'A
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Tid​

Answer 1

Step-by-step explanation:

Correct Question :-

Prove that ( Sin A+ Cosec A)^2+(Cos A + Sec A)^2 = 7+ Tan^2 A + Cot^2 A

Solution:-

LHS:-

( Sin A+ Cosec A)^2+(Cos A + Sec A)^2

We know that

(a+b)^2 = a^2+2ab+b^2

=> Sin^2 A+ 2 Sin A Cosec A+ Cosec^2 A+ Cos^2A+ Sec^2 A+ 2 Cos A Sec A

We know that

Sec A = 1/ Cos A

Sec A Cos A = 1

Cosec A = 1/ Sin A

Cosec A Sin A = 1

=> Sin^2 A+ Cosec^2 A + 2+Cos^2 A+ Sec^2 A+2

=>Sin^2 A+ Cosec^2 A +Cos^2 A+ Sec^2 A + 4

=> (Sin^2 A+ Cos^2 A)+Cosec^2 A + Sec^2 A+4

We know that

Sin^2 A+ Cos^2 A = 1

=> 1 + Cosec^2 A + Sec^2 Ab+4

=> 5 + Cosec^2 A + Sec^2 A

We know that

Sec^2 A - Tan^2 A = 1

Cosec^2 A - Cot^2 A = 1

=> 5 + 1+Tan^2 A + 1 + Cot^2 A

=> 7 + Tan^2 A + Cot^2 A

=> RHS

LHS = RHS

Hence, Proved.

Used formulae:-

• (a+b)^2 = a^2+2ab+b^2

• Sec A = 1/ Cos A
• Sec A Cos A = 1

• Cosec A = 1/ Sin A

• Cosec A Sin A = 1

• Sec^2 A - Tan^2 A = 1

• Cosec^2 A - Cot^2 A = 1