Advertisements
Advertisements
Question
Define an identity.
Solution
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
RELATED QUESTIONS
Prove the following identities:
`(i) (sinθ + cosecθ)^2 + (cosθ + secθ)^2 = 7 + tan^2 θ + cot^2 θ`
`(ii) (sinθ + secθ)^2 + (cosθ + cosecθ)^2 = (1 + secθ cosecθ)^2`
`(iii) sec^4 θ– sec^2 θ = tan^4 θ + tan^2 θ`
Show that `sqrt((1-cos A)/(1 + cos A)) = sinA/(1 + cosA)`
Prove the following trigonometric identities.
(1 + cot A − cosec A) (1 + tan A + sec A) = 2
Prove the following trigonometric identities.
`(cos A cosec A - sin A sec A)/(cos A + sin A) = cosec A - sec A`
Prove the following identities:
(1 + tan A + sec A) (1 + cot A – cosec A) = 2
Write the value of ` cosec^2 (90°- theta ) - tan^2 theta`
Write the value of `cosec^2 theta (1+ cos theta ) (1- cos theta).`
If 5x = sec ` theta and 5/x = tan theta , " find the value of 5 "( x^2 - 1/( x^2))`
If x = a sin θ and y = b cos θ, what is the value of b2x2 + a2y2?
What is the value of \[\frac{\tan^2 \theta - \sec^2 \theta}{\cot^2 \theta - {cosec}^2 \theta}\]
If \[sec\theta + tan\theta = x\] then \[tan\theta =\]
The value of \[\sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}}\]
Prove the following identity :
`1/(sinA + cosA) + 1/(sinA - cosA) = (2sinA)/(1 - 2cos^2A)`
If tanA + sinA = m and tanA - sinA = n , prove that (`m^2 - n^2)^2` = 16mn
Find the value of `θ(0^circ < θ < 90^circ)` if :
`cos 63^circ sec(90^circ - θ) = 1`
Prove the following identities: sec2 θ + cosec2 θ = sec2 θ cosec2 θ.
Choose the correct alternative:
sec2θ – tan2θ =?
Given that sinθ + 2cosθ = 1, then prove that 2sinθ – cosθ = 2.
If sinθ = `11/61`, then find the value of cosθ using the trigonometric identity.
`(cos^2 θ)/(sin^2 θ) - 1/(sin^2 θ)`, in simplified form, is ______.
