Prove each of the following identities : `(tan theta)/((1+ tan^(2) theta)^(2)) + (cot theta)/((1+ cot^(2) theta)^(2)) = sin theta cos theta `

Prove each of the following identities : `(tan theta)/((1+ tan^(2) the

1.	Prove each of the following identities : `(tan theta)/((1+ tan^(2) theta)^(2)) + (cot theta)/((1+ cot^(2) theta)^(2)) = sin theta cos theta `
Answer» LHS ` = (tan theta )/(sec^(4) theta ) + (cot theta )/("cosec"^(4) theta ) = ((sin theta)/(cos theta) xx cos^(4) theta ) + ((cos theta)/(sin theta) xx sin^(4) theta) ` ` = (sin theta cos^(3) theta + sin^(3) theta cos theta ) = sin theta cos theta ( cos^(2) theta + sin^(2) theta ) = sin theta cos theta . `

`1/(c o s e ctheta-cottheta)-1/(sintheta)=1/(sintheta)-1/(c o s e ctheta+cottheta)`
If ` x=r sin alpha cos beta, y= r sin alpha sin beta and z= r cos alpha , ` prove that ` x^(2)+y^(2) + z^(2) = r^(2). `
If `cos e ctheta-sintheta=ma n dsectheta-costheta=n ,`prove that `(m^(2n))^(2/3)+(m m^2)^(2/3)=1`
If `(3 sin theta + 5 cos theta ) = 5, ` prove that ` (5 sin theta - 3 cos theta ) = pm 3. `
If a `costheta-bsintheta=c ,`prove that a `sintheta+bcostheta=+-sqrt(a^2+b^2-c^2)`
Choose the correct answer in each of the following questions: `(sec 30^(@))/("cosec"60^(@))=?`A. `(2)/(sqrt(3))`B. `(sqrt(3))/(2)`C. `sqrt(3)`D. 1
`cos1^@xxcos2^@xxcos3^@xx...xxcos180^@=?`A. -1B. 1C. 0D. `(1)/(2)`
Choose the correct answer in each of the following questions: `sin47^(@)cos43^(@)+cos47^(@)sin43^(@)=?`A. `sin4^(@)`B. `cos4^(@)`C. 1D. 0
Prove that `(sin theta + "cosec" theta)^(2)+(cos theta + sec theta)^(2)=(7+ tan^(2) theta + cot^(2) theta).`
Prove that `(i) (1- sin theta)/(1+ sin theta)=(sec theta - tan theta )^(2) ` `(ii) ((1+cos theta))/((1- cos theta))= ("cosec" theta + cot theta)^(2)`