Advertisements
Advertisements
Question
If cos (\[\alpha + \beta\]= 0 , then sin \[\left( \alpha - \beta \right)\] can be reduced to
Solution
It is given that,
\[\cos\left( \alpha + \beta \right) = 0\]
\[ \Rightarrow \cos\left( \alpha + \beta \right) = \cos90° \left( \cos90° = 0 \right)\]
\[ \Rightarrow \alpha + \beta = 90° \]
\[ \Rightarrow \alpha = 90°- \beta\]
\[\text{ Now, put }\alpha = 90°- \beta in \sin\left( \alpha - \beta \right) . \]
\[ \therefore \sin\left( \alpha - \beta \right)\]
\[ = \sin\left( 90° - \beta - \beta \right)\]
\[ = \sin\left( 90°- 2\beta \right) \]
\[ = \cos2\beta \left[ \sin\left( 90° - \theta \right) = \cos\theta \right]\]
APPEARS IN
RELATED QUESTIONS
Prove the following trigonometric identities
`cos theta/(1 - sin theta) = (1 + sin theta)/cos theta`
Prove that `sqrt((1 + cos theta)/(1 - cos theta)) + sqrt((1 - cos theta)/(1 + cos theta)) = 2 cosec theta`
Prove the following identities:
cosec4 A – cosec2 A = cot4 A + cot2 A
Prove the following identities:
(cosec A – sin A) (sec A – cos A) (tan A + cot A) = 1
Prove the following identities:
`tan^2A - tan^2B = (sin^2A - sin^2B)/(cos^2A * cos^2B)`
Prove that:
`(tanA + 1/cosA)^2 + (tanA - 1/cosA)^2 = 2((1 + sin^2A)/(1 - sin^2A))`
Prove that:
(1 + tan A . tan B)2 + (tan A – tan B)2 = sec2 A sec2 B
If `sec theta + tan theta = p,` prove that
(i)`sec theta = 1/2 ( p+1/p) (ii) tan theta = 1/2 ( p- 1/p) (iii) sin theta = (p^2 -1)/(p^2+1)`
If m = ` ( cos theta - sin theta ) and n = ( cos theta + sin theta ) "then show that" sqrt(m/n) + sqrt(n/m) = 2/sqrt(1-tan^2 theta)`.
If `tan theta = 1/sqrt(5), "write the value of" (( cosec^2 theta - sec^2 theta))/(( cosec^2 theta - sec^2 theta))`
Four alternative answers for the following question are given. Choose the correct alternative and write its alphabet:
sin θ × cosec θ = ______
If cos A + cos2 A = 1, then sin2 A + sin4 A =
If a cos θ − b sin θ = c, then a sin θ + b cos θ =
Prove the following identity :
`sec^2A.cosec^2A = tan^2A + cot^2A + 2`
Prove the following identity :
`cosA/(1 - tanA) + sinA/(1 - cotA) = sinA + cosA`
Prove the following identity :
`(cotA + cosecA - 1)/(cotA - cosecA + 1) = (cosA + 1)/sinA`
Without using trigonometric table, prove that
`cos^2 26° + cos 64° sin 26° + (tan 36°)/(cot 54°) = 2`
Prove that: sin6θ + cos6θ = 1 - 3sin2θ cos2θ.
Prove that sin θ (1 – tan θ) – cos θ (1 – cot θ) = cosec θ – sec θ
Prove that `sqrt(sec^2 theta + "cosec"^2 theta) = tan theta + cot theta`
