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प्रश्न
Prove the following trigonometric identities.
`cos A/(1 - tan A) + sin A/(1 - cot A) = sin A + cos A`
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उत्तर
We need to prove `cos A/(1 - tan A) + sin A/(1 - cot A) = sin A + cos `
Solving the L.H.S, we get
`cos A/(1 - tan A) + sin A/(1 - cot A)`
= `cos A/(1 - sin A/cos A) + sin A/(1 - cos A/sin A)`
`= cos A/((cos A - sin A)/cos A) + sin A/((sin A - cos A)/sin A)`
`= cos^2 A/(cos A - sin A) + (sin^2 A)/(sin A - cos A)`
`= (cos^2 A - sin^2 A)/(cos A - sin A)`
`= ((cos A + sin A)(cos A - sin A))/(cos A - sin A)` [using `a^2 - b^2 = (a + b)(a -b)`]
= cos A + sin A
= RHS
Hence proved.
संबंधित प्रश्न
Prove the following identities:
`( i)sin^{2}A/cos^{2}A+\cos^{2}A/sin^{2}A=\frac{1}{sin^{2}Acos^{2}A)-2`
`(ii)\frac{cosA}{1tanA}+\sin^{2}A/(sinAcosA)=\sin A\text{}+\cos A`
`( iii)((1+sin\theta )^{2}+(1sin\theta)^{2})/cos^{2}\theta =2( \frac{1+sin^{2}\theta}{1-sin^{2}\theta } )`
If x = r cos A cos B, y = r cos A sin B and z = r sin A, show that : x2 + y2 + z2 = r2
Prove the following identities:
`(1 + (secA - tanA)^2)/(cosecA(secA - tanA)) = 2tanA`
Write the value of `( 1- sin ^2 theta ) sec^2 theta.`
If 5x = sec θ and \[\frac{5}{x} = \tan \theta\]find the value of \[5\left( x^2 - \frac{1}{x^2} \right)\]
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If tanA + sinA = m and tanA - sinA = n , prove that (`m^2 - n^2)^2` = 16mn
Verify that the points A(–2, 2), B(2, 2) and C(2, 7) are the vertices of a right-angled triangle.
Prove that cos2θ . (1 + tan2θ) = 1. Complete the activity given below.
Activity:
L.H.S. = `square`
= `cos^2θ xx square` ...`[1 + tan^2θ = square]`
= `(cos θ xx square)^2`
= 12
= 1
= R.H.S.
`5/(sin^2θ) - 5cot^2θ`, complete the activity given below.
Activity:
`5/(sin^2θ) - 5cot^2θ`
= `square (1/(sin^2θ) - cot^2θ)`
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= 5(1)
= `square`
