RS Aggarwal solutions for Mathematics [English] Class 10 chapter 8 - Trigonometric Identities [Latest edition]

(i)` (1-cos^2 theta )cosec^2theta = 1`

`(1 + cot^2 theta ) sin^2 theta =1`

`(sec^2 theta-1) cot ^2 theta=1`

`(sec^2 theta -1)(cosec^2 theta - 1)=1`

`(1-cos^2theta) sec^2 theta = tan^2 theta`

`sin^2 theta + 1/((1+tan^2 theta))=1`

`1/((1+tan^2 theta)) + 1/((1+ tan^2 theta))`

`(1+ cos theta)(1- costheta )(1+cos^2 theta)=1`

`cosec theta (1+costheta)(cosectheta - cot theta )=1`

`cot^2 theta - 1/(sin^2 theta ) = -1`a

` tan^2 theta - 1/( cos^2 theta )=-1`

`cos^2 theta + 1/((1+ cot^2 theta )) =1`

`1/((1+ sintheta ))+1/((1- sin theta ))= 2 sec^2 theta`

`sec theta (1- sin theta )( sec theta + tan theta )=1`

`sin theta (1+ tan theta) + cos theta (1+ cot theta) = ( sectheta+ cosec  theta)`

`1+ (cot^2 theta)/((1+ cosec theta))= cosec theta`

`1+(tan^2 theta)/((1+ sec theta))= sec theta`

`1+((tan^2 theta) cot theta)/(cosec^2 theta) = tan theta`

`(tan^2theta)/((1+ tan^2 theta))+ cot^2 theta/((1+ cot^2 theta))=1`

If sinθ – cosθ = 0, then the value of (sin⁴θ + cos⁴θ) is ______.

`tan theta /((1 - cot theta )) + cot theta /((1 - tan theta)) = (1+ sec theta cosec  theta)`

`cos^2 theta /((1 tan theta))+ sin ^3 theta/((sin theta - cos theta))=(1+sin theta cos theta)`

`costheta/((1-tan theta))+sin^2theta/((cos theta-sintheta))=(cos theta+ sin theta)`

`(1+tan^2theta)(1+cot^2 theta)=1/((sin^2 theta- sin^4theta))`

`tan theta/(1+ tan^2 theta)^2 + cottheta/(1+ cot^2 theta)^2 = sin theta cos theta`

`sin^6 theta + cos^6 theta =1 -3 sin^2 theta cos^2 theta`

`sin^2 theta + cos^4 theta = cos^2 theta + sin^4 theta`

cosec⁴θ − cosec²θ = cot⁴θ + cot²θ

`(1+ tan^2 theta)/(1+ tan^2 theta)= (cos^2 theta - sin^2 theta)`

`(1-tan^2 theta)/(cot^2-1) = tan^2 theta`

`(tan theta)/((sec theta -1))+(tan theta)/((sec theta +1)) = 2 sec theta`

`cot theta/((cosec  theta + 1) )+ ((cosec  theta +1 ))/ cot theta = 2 sec theta `

`(sec theta -1 )/( sec theta +1) = ( sin ^2 theta)/( (1+ cos theta )^2)`

`(sectheta- tan theta)/(sec theta + tan theta) = ( cos ^2 theta)/( (1+ sin theta)^2)`

`sqrt((1+sin theta)/(1-sin theta)) = (sec theta + tan theta)`

`sqrt((1-cos theta)/(1+cos theta)) = (cosec  theta - cot  theta)`

`sqrt((1+cos theta)/(1-cos theta)) + sqrt((1-cos theta )/(1+ cos theta )) = 2 cosec theta`

`(cos^3 theta +sin^3 theta)/(cos theta + sin theta) + (cos ^3 theta - sin^3 theta)/(cos theta - sin theta) = 2`

`sin theta/((cot theta + cosec  theta)) - sin theta /( (cot theta - cosec  theta)) =2`

` (sin theta - cos theta) / ( sin theta + cos theta ) + ( sin theta + cos theta ) / ( sin theta - cos theta ) = 2/ ((2 sin^2 theta -1))`

` (sin theta + cos theta )/(sin theta - cos theta ) + ( sin theta - cos theta )/( sin theta + cos theta) = 2/ ((1- 2 cos^2 theta))`

`(1+ cos  theta - sin^2 theta )/(sin theta (1+ cos theta))= cot theta`

`(cos  ec^theta + cot theta )/( cos ec theta - cot theta  ) = (cosec theta + cot theta )^2 = 1+2 cot^2 theta + 2cosec theta  cot theta`

`(sec theta + tan theta )/( sec theta - tan theta ) = ( sec theta + tan theta )^2 = 1+2 tan^2 theta + 25 sec theta tan theta `

`(1+ cos theta + sin theta)/( 1+ cos theta - sin theta )= (1+ sin theta )/(cos theta)`

`(sin theta+1-cos theta)/(cos theta-1+sin theta) = (1+ sin theta)/(cos theta)`

`(sin theta)/((sec theta + tan theta -1)) + cos theta/((cosec theta + cot theta -1))=1`

`(sin theta +cos theta )/(sin theta - cos theta)+(sin theta- cos theta)/(sin theta + cos theta) = 2/((sin^2 theta - cos ^2 theta)) = 2/((2 sin^2 theta -1))`

`(cos theta  cosec theta - sin theta sec theta )/(costheta + sin theta) = cosec theta - sec theta`

`(1+ tan theta + cot theta )(sintheta - cos theta) = ((sec theta)/ (cosec^2 theta)-( cosec theta)/(sec^2 theta))`

`(cot^2 theta ( sec theta - 1))/((1+ sin theta))+ (sec^2 theta(sin theta-1))/((1+ sec theta))=0`

`{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)`

`((sin A-  sin B ))/(( cos A + cos B ))+ (( cos A - cos B ))/(( sinA + sin B ))=0` 

`(tan A + tanB )/(cot A + cot B) = tan A tan B`

Show that none of the following is an identity:
(i) `cos^2theta + cos theta =1`

Show that none of the following is an identity: 

`sin^2 theta + sin  theta =2`

Show that none of the following is an identity:

`tan^2 theta + sin theta = cos^2 theta`

Prove that `( sintheta - 2 sin ^3 theta ) = ( 2 cos ^3 theta - cos theta) tan theta`

If a cos `theta + b sin theta = m and a sin theta - b cos theta = n , "prove that "( m^2 + n^2 ) = ( a^2 + b^2 )`

If x= a sec `theta + b tan theta and y = a tan theta + b sec theta ,"prove that" (x^2 - y^2 )=(a^2 -b^2)`

If `(x/a sin a - y/b cos theta) = 1 and (x/a cos theta + y/b sin theta ) =1, " prove that "(x^2/a^2 + y^2/b^2 ) =2`

If` (sec theta + tan theta)= m and ( sec theta - tan theta ) = n ,` show that mn =1

If `( cosec theta + cot theta ) =m and ( cosec theta - cot theta ) = n, ` show that mn = 1.

If x=a `cos^3 theta and y = b sin ^3 theta ," prove that " (x/a)^(2/3) + ( y/b)^(2/3) = 1.`

If `( tan theta + sin theta ) = m and ( tan theta - sin theta ) = n " prove that "(m^2-n^2)^2 = 16 mn .`

If `(cot theta ) = m and ( sec theta - cos theta) = n " prove that " (m^2 n)(2/3) - (mn^2)(2/3)=1`

If `(cosec theta - sin theta )= a^3 and (sec theta - cos theta ) = b^3 , " prove that " a^2 b^2 ( a^2+ b^2 ) =1`

If`( 2 sin theta + 3 cos theta) =2 , " prove that " (3 sin theta - 2 cos theta) = +- 3.`

If `( sin theta + cos theta ) = sqrt(2) , " prove that " cot theta = ( sqrt(2)+1)`.

If `( cos theta + sin theta) = sqrt(2) sin theta , " prove that " ( sin theta - cos theta ) = sqrt(2) cos theta`

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 tan A = n tan B and sin A = m sin B , prove that  `cos^2 A = ((m^2-1))/((n^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)`.

Write the value of `( 1- sin ^2 theta  ) sec^2 theta.`

Write the value of `(1 - cos^2 theta ) cosec^2 theta`.

Write the value of `(1 + tan^2 theta ) cos^2 theta`. 

Write the value of `(1 + cot^2 theta ) sin^2 theta`. 

Write the value of `(sin^2 theta 1/(1+tan^2 theta))`. 

Write the value of `(cot^2 theta -  1/(sin^2 theta))`. 

Write the value of `sin theta cos ( 90° - theta )+ cos theta sin ( 90° - theta )`. 

Write the value of ` cosec^2 (90°- theta ) - tan^2 theta`

Write the value of ` sec^2 theta ( 1+ sintheta )(1- sintheta).`

Write the value of `cosec^2 theta (1+ cos theta ) (1- cos theta).`

Write the value of ` sin^2 theta cos^2 theta (1+ tan^2 theta ) (1+ cot^2 theta).`

Write the value of `(1+ tan^2 theta ) ( 1+ sin theta ) ( 1- sin theta)`

Write the value of `3 cot^2 theta - 3 cosec^2 theta.`

Write the value of `4 tan^2 theta  - 4/ cos^2 theta`

Write the value of`(tan^2 theta  - sec^2 theta)/(cot^2 theta - cosec^2 theta)`

If  `sin theta = 1/2 , " write the value of" ( 3 cot^2 theta + 3).`

If `cos theta = 2/3 , " write the value of" (4+4 tan^2 theta).`

If `cos theta = 7/25 , "write the value of" ( tan theta + cot theta).`

If `cos theta = 2/3 , "write the value of" ((sec theta -1))/((sec theta +1))`

If 5 `tan theta = 4,"write the value of" ((cos theta - sintheta))/(( cos theta + sin theta))`

If 3 `cot theta = 4 , "write the value of" ((2 cos theta - sin theta))/(( 4 cos theta - sin theta))`

If `cot theta = 1/ sqrt(3) , "write the value of" ((1- cos^2 theta))/((2 -sin^2 theta))`

If `tan theta = 1/sqrt(5), "write the value of" (( cosec^2 theta - sec^2 theta))/(( cosec^2 theta - sec^2 theta))`

If ` cot A= 4/3 and (A+ B) = 90°  `  ,what is the value of tan B?

If `cos B = 3/5 and (A + B) =- 90° ,`find the value of sin A.

If `sqrt(3) sin theta = cos theta  and theta ` is an acute angle, find the value of θ .

Write the value of tan10° tan 20° tan 70° tan 80° .

Write the value of tan1° tan 2°   ........ tan 89° .

Write the value of cos1° cos 2°........cos180° .

If tan A =` 5/12` ,  find the value of (sin A+ cos A) sec A.

`If sin theta = cos( theta - 45° ),where   theta   " is   acute, find the value of "theta` .

Find the value of ` ( sin 50°)/(cos 40°)+ (cosec 40°)/(sec 50°) - 4 cos 50°   cosec 40 °`

Find the value of sin ` 48° sec 42° + cos 48°  cosec 42°`

If x =  a sin θ and y = bcos θ , write the value of`(b^2 x^2 + a^2 y^2)`

If 5x = sec ` theta and 5/x = tan theta , " find the value of 5 "( x^2 - 1/( x^2))`

If `cosec  theta = 2x and cot theta = 2/x ," find the value of"  2 ( x^2 - 1/ (x^2))`

If `sec theta + tan theta = x,"  find the value of " sec theta`

Find the value of `(cos 38° cosec 52°)/(tan 18° tan 35° tan 60° tan 72° tan 55°)`

If `sin theta = x , " write the value of cot "theta .`

If `sec theta = x ,"write the value of tan"  theta`.