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# R.S. Aggarwal solutions for Class 10 Mathematics chapter 8 - Trigonometric Identities

## Secondary School Mathematics for Class 10 (for 2019 Examination)

#### R.S. Aggarwal Secondary School Mathematics Class 10 (for 2019 Examination) ## Chapter 8: Trigonometric Identities

#### Chapter 8: Trigonometric Identities solutions [Page 0]

(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 ) = -1a

 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

sin theta / ((1+costheta))+((1+costheta))/sin theta=2cosectheta

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

cosec ^4 theta + cosec^2 theta = cot^4 theta+ cot^2 theta

(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

#### Chapter 8: Trigonometric Identities solutions [Page 0]

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).

#### Chapter 8: Trigonometric Identities solutions [Page 0]

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.

## Chapter 8: Trigonometric Identities

#### R.S. Aggarwal Secondary School Mathematics Class 10 (for 2019 Examination) ## R.S. Aggarwal solutions for Class 10 Mathematics chapter 8 - Trigonometric Identities

R.S. Aggarwal solutions for Class 10 Maths chapter 8 (Trigonometric Identities) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Secondary School Mathematics for Class 10 (for 2019 Examination) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 10 Mathematics chapter 8 Trigonometric Identities are Trigonometric Ratios of Complementary Angles, Trigonometric Identities.

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