Question

Prove the following 1/1+ sin^2 + 1/1+ cos^2+ 1/ sec2 + 1/ cosec^2=2

Answer 1

here is your answer....

Answer 2

Answer:

According to the equation,

We have been provided that,

$\frac{1}{1+sin^{2}x}+\frac{1}{1+cos^{2}x}+\frac{1}{1+sec^{2}x}+\frac{1}{1+cosec^{2}x}=2$

We need to prove the given statement,

Now,

Using the Trigonometric identities, we can say that,

sec²x = 1 / cos²x

and,

cosec²x = 1 / sin²x

Therefore, on putting the required values in the given equation, we get,

$\frac{1}{1+sin^{2}x}+\frac{1}{1+cos^{2}x}+\frac{1}{1+sec^{2}x}+\frac{1}{1+cosec^{2}x}=2\\\frac{1}{1+sin^{2}x}+\frac{1}{1+cos^{2}x}+\frac{1}{1+\frac{1}{cos^{2}x}}+\frac{1}{1+\frac{1}{sin^{2}x}}=2$

Therefore, on simplifying it further by taking the terms in the numerator we get,

$\frac{1}{1+sin^{2}x}+\frac{1}{1+cos^{2}x}+\frac{1}{1+\frac{1}{cos^{2}x}}+\frac{1}{1+\frac{1}{sin^{2}x}}=2\\\frac{1}{1+sin^{2}x}+\frac{1}{1+cos^{2}x}+\frac{cos^{2}x}{1+cos^{2}x}+\frac{sin^{2}x}{1+sin^{2}x}=2\\\frac{1}{1+sin^{2}x}+\frac{1+cos^{2}x}{1+cos^{2}x}+\frac{sin^{2}x}{1+sin^{2}x}=2\\\frac{1+sin^{2}x}{1+sin^{2}x}+\frac{1+cos^{2}x}{1+cos^{2}x}=2\\1 +1=2\\2=2$

Therefore, we can see in the end that,

LHS = RHS

Hence, Proved.