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Question
Prove the following trigonometric identities.
`(1/(sec^2 theta - cos theta) + 1/(cosec^2 theta - sin^2 theta)) sin^2 theta cos^2 theta = (1 - sin^2 theta cos^2 theta)/(2 + sin^2 theta + cos^2 theta)`
Solution
In the given question, we need to prove
`(1/(sec^2 theta - cos theta) + 1/(cosec^2 theta - sin^2 theta)) sin^2 theta cos^2 theta = (1 - sin^2 theta cos^2 theta)/(2 + sin^2 theta + cos^2 theta)`
Now using `sec theta = 1/ cos theta` and `cosec theta = 1/sin theta` in LHS we get
LHS =`(1/((1/cos^2 theta) - cos^2 theta) + 1/((1/sin^2 theta) - sin^2 theta)) sin^2 theta cos^2 theta`
`= (1/((1 - cos^4 theta)/cos^2 theta) + 1/((1 - sin^4 theta)/sin^2 theta)) sin^2 theta cos^2 theta`
`= ((cos^2 theta)/(1 - cos^4 theta) + sin^2 theta/(1 - sin^4 theta)) sin^2 theta cos^2 theta`
Further using the identity `a^2 - b^2 = (a + b)(a- b)` we get
LHS = `(cos^2 theta/((1 - cos^2 theta)(1 + cos^2 theta)) + sin^2 theta/((1 - sin^2 theta) (1 + sin^2 theta)))sin^2 theta cos^2 theta`
`= ((cos^2 theta)/(sin^2 theta(1 + cos^2 theta)) + sin^2 theta/(cos^2 theta(1 + sin^2 theta))) sin^2 theta cos^2 theta`
`= ((cos^2 theta(cos^2 theta(1 + sin^2 theta))+sin^2 theta(sin^2 theta(1 + cos^2 theta)))/(sin^2 theta cos^2 theta(1 + cos^2 theta)(1 +sin^2 theta))) sin^2 theta cos^2 theta`
`= ((cos^4 theta(1 + sin^2 theta) + sin^4 theta(1 + cos^2 theta))/((1 + cos^2 theta)(1 + sin^2 theta)))`
Further using the identity `sin^2 theta + cos^2 theta = 1` we get
LHS = `((cos^4 theta + cos^4 theta sin^2 theta + sin^4 theta + sin^4 theta cos^2 theta)/(1 + cos^2 theta + sin^2 theta + sin^2 theta cos^2 theta))`
`= (cos^4 theta + sin^4 theta + cos^2 theta sin^2 theta (cos^2 theta + sin^2 theta)) /(2 + sin^2 theta cos^2theta)`
`= ((cos^4 theta +sin^4 theta +cos^2 theta sin^2theta (1))/(2 + sin^2 theta cos^2 theta))`
Now, from the identity `a^2 + b^2 = (a + b)^2 - 2ab` we get
So,
LHS = `(((cos^2 theta + sin^2 theta)^2 - 2cos^2 theta sin^2 theta +cos^2 theta sin^2 theta)/(2 + sin^2 theta cos^2 theta))`
`= (((1)^2 - cos^2 theta sin^2 theta)/(22 +sin^2 theta cos^2 theta))`
`= ((1 - sin^2 theta cos^2 theta)/(2 + sin^2 theta cos^2 theta))`
Hence proved.
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