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Please help me thank you

Please help me thank you-example-1
User Csvan
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1 Answer

4 votes

Answer:

option C Only second equation is an identity is correct.

Explanation:

1)


1 +(cos^2\theta)/(cot^2\theta(1-sin^2\theta))= 9 sec^2\theta

We need to prove this identity.

We know:


cos^2\theta + sin^2\theta = 1\\=> cos^2\theta = 1- sin^2\theta

and


(1)/(cot^2\theta ) = tan^2\theta \\and\\tan^2\theta = (sin^2\theta)/(cos^2\theta)

Using these to solve the identity


1 +(cos^2\theta)/(cot^2\theta(cos^2\theta))= 9 sec^2\theta\\1 +(1)/(cot^2\theta) = 9 sec^2\theta\\1+tan^2\theta = 9 sec^2\theta\\1+(sin^2\theta)/(cos^2\theta) = 9sec^2\theta\\\\(cos^2\theta+sin^2\theta)/(cos^2\theta) = 9sec^2\theta\\(1)/(cos^2\theta)=9sec^2\theta \\sec^2\theta \\eq 9sec^2\theta

So, this is not an identity.

2)


20sin\theta((1)/(sin\theta) -(cot\theta)/(sec\theta)) =20sin^2\theta\\

We need to prove this identity.

We know:


cot\theta = (cos\theta)/(sin\theta) \\and \\sec\theta=(1)/(cos\theta) \\so,\,\, (cot\theta)/(sec\theta)= ((cos\theta)/(sin\theta))/((1)/(cos\theta))  \\Solving\\ (cot\theta)/(sec\theta) =(cos^2\theta)/(sin\theta)

Using this to solve the identity


20sin\theta((1)/(sin\theta) -(cot\theta)/(sec\theta)) =20sin^2\theta\\\\Putting\,\,values\,\,\\ 20sin\theta((1)/(sin\theta) -(cos^2\theta)/(sin\theta)) =20sin^2\theta\\ 20sin\theta((1-cos^2\theta)/(sin\theta)) =20sin^2\theta\\1-cos^2\theta = sin^2\theta\\ 20sin\theta((sin^2\theta)/(sin\theta)) =20sin^2\theta\\Cancelling\,\, sin\theta \,\,over \,\,sin\theta\\20sin^2\theta=20sin^2\theta

So, this is an identity.

So, option C Only second equation is an identity is correct.

User Allon Guralnek
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