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प्रश्न
Define an identity.
उत्तर
An identity is an equation which is true for all values of the variable (s).
For example,
`(x+3)^2=x^2+6x+9`
Any number of variables may involve in an identity.
An example of an identity containing two variables is
`(x+y)^2=x^2+2xy+y^2`
The above are all about algebraic identities. Now, we define the trigonometric identities.
An equation involving trigonometric ratios of an angle 0 (say) is said to be a trigonometric identity if it is satisfied for all valued of 0 for which the trigonometric ratios are defined.
For examples,
\[\sin^2 \theta + \cos^2 \theta = 1\]
\[1 + \tan^2 \theta = \sec^2 \theta\]
\[1 + \cot^2 \theta = {cosec}^2 \theta\]
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संबंधित प्रश्न
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`(sec theta + tan theta )/( sec theta - tan theta ) = ( sec theta + tan theta )^2 = 1+2 tan^2 theta + 25 sec theta tan theta `
`(1+ tan theta + cot theta )(sintheta - cos theta) = ((sec theta)/ (cosec^2 theta)-( cosec theta)/(sec^2 theta))`
If `sqrt(3) sin theta = cos theta and theta ` is an acute angle, find the value of θ .
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If cosec θ − cot θ = α, write the value of cosec θ + cot α.
Write True' or False' and justify your answer the following :
The value of \[\sin \theta\] is \[x + \frac{1}{x}\] where 'x' is a positive real number .
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`(secA - 1)/(secA + 1) = (1 - cosA)/(1 + cosA)`
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If tan α = n tan β, sin α = m sin β, prove that cos2 α = `(m^2 - 1)/(n^2 - 1)`.
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Prove that: `(sin θ - 2sin^3 θ)/(2 cos^3 θ - cos θ) = tan θ`.
If cot θ = `40/9`, find the values of cosec θ and sinθ,
We have, 1 + cot2θ = cosec2θ
1 + `square` = cosec2θ
1 + `square` = cosec2θ
`(square + square)/square` = cosec2θ
`square/square` = cosec2θ ......[Taking root on the both side]
cosec θ = `41/9`
and sin θ = `1/("cosec" θ)`
sin θ = `1/square`
∴ sin θ = `9/41`
The value is cosec θ = `41/9`, and sin θ = `9/41`
