Advertisements
Advertisements
प्रश्न
Define an identity.
Advertisements
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,
\[\sin^2 \theta + \cos^2 \theta = 1\]
\[1 + \tan^2 \theta = \sec^2 \theta\]
\[1 + \cot^2 \theta = {cosec}^2 \theta\]
APPEARS IN
संबंधित प्रश्न
(1 + tan θ + sec θ) (1 + cot θ − cosec θ) = ______.
Prove that `(sin theta)/(1-cottheta) + (cos theta)/(1 - tan theta) = cos theta + sin theta`
Prove the following trigonometric identities.
`(tan^2 A)/(1 + tan^2 A) + (cot^2 A)/(1 + cot^2 A) = 1`
Prove the following identities:
`(cosecA - 1)/(cosecA + 1) = (cosA/(1 + sinA))^2`
(i)` (1-cos^2 theta )cosec^2theta = 1`
`{1/((sec^2 theta- cos^2 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)`
Show that none of the following is an identity:
`tan^2 theta + sin theta = cos^2 theta`
Prove that :(sinθ+cosecθ)2+(cosθ+ secθ)2 = 7 + tan2 θ+cot2 θ.
Prove that: 2(sin6 θ + cos6 θ) – 3 (sin4 θ + cos4 θ) + 1 = 0.
If x = r sin θ cos Φ, y = r sin θ sin Φ and z = r cos θ, prove that x2 + y2 + z2 = r2.
Prove that sin4θ - cos4θ = sin2θ - cos2θ
= 2sin2θ - 1
= 1 - 2 cos2θ
Prove that ( 1 + tan A)2 + (1 - tan A)2 = 2 sec2A
Prove that: 2(sin6θ + cos6θ) - 3 ( sin4θ + cos4θ) + 1 = 0.
Prove that `( 1 + sin θ)/(1 - sin θ) = 1 + 2 tan θ/cos θ + 2 tan^2 θ` .
If A = 30°, verify that `sin 2A = (2 tan A)/(1 + tan^2 A)`.
Prove that (cosec A - sin A)( sec A - cos A) sec2 A = tan A.
If a cos θ – b sin θ = c, then prove that (a sin θ + b cos θ) = `± sqrt(a^2 + b^2 - c^2)`
`sin θ = 1/2`, then θ = ?
cos 45° = ?
Prove that `(1 + sin θ)/(1 - sin θ) = (sec θ + tan θ)^2`.
