Prove the following: `(cos (pi + x) cos (-x))/(sin(pi - x) cos (pi/2 + x)) = cot^2 x`

L.H.S. = `(cos (pi + x) cos (-x))/(sin(pi - x) cos (pi/2 + x))` Now using `sin (π - x) = sin x, cos (pi/2+x) = - sin x` L.H.S. = `(- cosx xx cosx )/(sinx (- sinxx)` = `cos^2x/sin^2x = (cosx/sinx)^2` = `cot^2xx` = R.H.S.

Prove the following: cos(π+x)cos(-x)sin(π-x)cos(π2+x)= cot2x - Mathematics

प्रश्न

Prove the following:

$\frac{\cos (π + x) \cos (- x)}{\sin (π - x) \cos (\frac{π}{2} + x)} = \cot^{2} x$

योग

उत्तर

L.H.S. = $\frac{\cos (π + x) \cos (- x)}{\sin (π - x) \cos (\frac{π}{2} + x)}$

Now using $\sin (π - x) = \sin x, \cos (\frac{π}{2} + x) = - \sin x$

L.H.S. = $\frac{- \cos x \times \cos x}{\sin x (- \sin \times)}$

= $\frac{\cos^{2} x}{\sin^{2} x} = {(\frac{\cos x}{\sin x})}^{2}$

= $\cot^{2} \times$

= R.H.S.

shaalaa.com

Trigonometric Functions of Sum and Difference of Two Angles

क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

Q 8 Q 7 Q 9

अध्याय 3: Trigonometric Functions - Exercise 3.3 [पृष्ठ ७३]

APPEARS IN

एनसीईआरटी Mathematics [English] Class 11

अध्याय 3 Trigonometric Functions
Exercise 3.3 | Q 8 | पृष्ठ ७३

वीडियो ट्यूटोरियलVIEW ALL [1]

view
वीडियो ट्यूटोरियल For All Subjects
Trigonometric Functions of Sum and Difference of Two Angles

video tutorial00:06:30

संबंधित प्रश्न

Prove that $\cot^{2} \frac{π}{6} + \cos e c \frac{5 π}{6} + 3 \tan^{2} \frac{π}{6} = 6$

Find the value of: tan 15°

Prove the following:

sin² 6x – sin² 4x = sin 2x sin 10x

Prove the following:

cos² 2x – cos² 6x = sin 4x sin 8x

Prove the following:

$\frac{\sin 5 x + \sin 3 x}{\cos 5 x + \cos 3 x} = \tan 4 x$

Prove the following:

$\frac{\cos 4 x + \cos 3 x + \cos 2 x}{\sin 4 x + \sin 3 x + \sin 2 x} = \cot 3 x$

Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x

If $\sin A = \frac{4}{5}$ and $\cos B = \frac{5}{13}$ , where 0 < A, $B < \frac{π}{2}$ , find the value of the following:
sin (A − B)

If $\sin A = \frac{1}{2}, \cos B = \frac{12}{13}$ , where $\frac{π}{2}$ < A < π and $\frac{3 π}{2}$ < B < 2π, find tan (A − B).

If $\cos A = - \frac{12}{13} and \cot B = \frac{24}{7}$ , where A lies in the second quadrant and B in the third quadrant, find the values of the following:
sin (A + B)

Prove that

\frac{\cos 9^{\circ} + \sin 9^{\circ}}{\cos 9^{\circ} - \sin 9^{\circ}} = \tan 54^{\circ}

Prove that:
sin² (n + 1) A − sin² nA = sin (2n + 1) A sin A.

Prove that:
cos² A + cos² B − 2 cos A cos B cos (A + B) = sin² (A + B)

Prove that:
tan 36° + tan 9° + tan 36° tan 9° = 1

Prove that sin² (n + 1) A − sin² nA = sin (2n + 1) A sin A.

If sin α + sin β = a and cos α + cos β = b, show that

\sin (α + β) = \frac{2 a b}{a^{2} + b^{2}}

If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x².

If angle $θ$ is divided into two parts such that the tangents of one part is $λ$ times the tangent of other, and $ϕ$ is their difference, then show that $\sin θ = \frac{λ + 1}{λ - 1} \sin ϕ$

Find the maximum and minimum values of each of the following trigonometrical expression:

$5 \cos x + 3 \sin (\frac{π}{6} - x) + 4$

Reduce each of the following expressions to the sine and cosine of a single expression:

$\sqrt{3} \sin x - \cos x$

Reduce each of the following expressions to the sine and cosine of a single expression:

24 cos x + 7 sin x

Prove that $(2 \sqrt{3} + 3) \sin x + 2 \sqrt{3} \cos x$ lies between $- (2 \sqrt{3} + \sqrt{15}) and (2 \sqrt{3} + \sqrt{15})$

Write the maximum and minimum values of 3 cos x + 4 sin x + 5.

If A + B + C = π, then sec A (cos B cos C − sin B sin C) is equal to

If in ∆ABC, tan A + tan B + tan C = 6, then cot A cot B cot C =

\frac{\cos 10^{\circ} + \sin 10^{\circ}}{\cos 10^{\circ} - \sin 10^{\circ}} =

If tan (π/4 + x) + tan (π/4 − x) = a, then tan² (π/4 + x) + tan² (π/4 − x) =

The maximum value of $\sin^{2} (\frac{2 π}{3} + x) + \sin^{2} (\frac{2 π}{3} - x)$ is

If cos (A − B) $= \frac{3}{5}$ and tan A tan B = 2, then

Match each item given under column C₁ to its correct answer given under column C₂.

C₁	C₂
(a) $\frac{1 - \cos x}{\sin x}$	(i) $\cot^{2} \frac{x}{2}$
(b) $\frac{1 + \cos x}{1 - \cos x}$	(ii) $\cot \frac{x}{2}$
(c) $\frac{1 + \cos x}{\sin x}$	(iii) $\| \cos x + \sin x \|$
(d) $\sqrt{1 + \sin 2 x}$	(iv) $\tan \frac{x}{2}$

If $\frac{\sin (x + y)}{\sin (x - y)} = \frac{a + b}{a - b}$ , then show that $\frac{\tan x}{\tan y} = \frac{a}{b}$ [Hint: Use Componendo and Dividendo].

If sinθ + cosθ = 1, then find the general value of θ.

If sinθ + cosecθ = 2, then sin²θ + cosec²θ is equal to ______.

If α + β = $\frac{π}{4}$ , then the value of (1 + tan α)(1 + tan β) is ______.

State whether the statement is True or False? Also give justification.

If cosecx = 1 + cotx then x = 2nπ, 2nπ + $\frac{π}{2}$

State whether the statement is True or False? Also give justification.

If tanθ + tan2θ + $\sqrt{3}$ tanθ tan2θ = $\sqrt{3}$ , then θ = $\frac{n π}{3} + \frac{π}{9}$

Prove the following: cos(π+x)cos(-x)sin(π-x)cos(π2+x)= cot2x - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तर

APPEARS IN

वीडियो ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्न

Select a course

Prove the following: cos(π+x)cos(-x)sin(π-x)cos(π2+x)= cot2x - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तरShow Solution

APPEARS IN

वीडियो ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्न

Select a course

उत्तर