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RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1

March 4, 2019 by Veer Leave a Comment

RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 is part of RBSE Solutions for Class 10 Maths. Here we have given Rajasthan Board RBSE Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Exercise 7.1.

Board RBSE
Textbook SIERT, Rajasthan
Class Class 10
Subject Maths
Chapter Chapter 7
Chapter Name त्रिकोणमितीय सर्वसमिकाएँ
Exercise Exercise 7.1
Number of Questions Solved 30
Category RBSE Solutions

Rajasthan Board RBSE Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1

प्रश्न 1.
∠θ के लिए सभी त्रिकोणमितीय अनुपातों को sec θ के पदों में (RBSESolutions.com) व्यक्त कीजिए।
हल:
हम जानते हैं कि
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 1
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 2

RBSE Solutions

प्रश्न 2.
त्रिकोणमितीय अनुपातों sin θ, sec θ, tan θ को cot θ के पदों में (RBSESolutions.com) व्यक्त कीजिए।
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 3

निम्नलिखित को सर्वसमिकाओं की सहायता से सिद्ध कीजिए

प्रश्न 3.
cos2θ + cos2θ . cot2θ = cot2θ
हल:
L.H.S. = cos2θ + cos2θ . cot2θ
= cos2θ6(1 + cot2θ)
= cos2θ . cosec2θ [∵ 1 + cot2θ = cosec2θ]
\(=\cos ^{2} \theta \cdot \frac{1}{\sin ^{2} \theta} \quad\left[\because \csc \theta=\frac{1}{\sin \theta}\right]\)
\(=\frac{\cos ^{2} \theta}{\sin ^{2} \theta}=\cot ^{2} \theta\) = R.H.S
∴ L.H.S. = R.H.S. ( इतिसिद्धम् )

प्रश्न 4.
sec θ(1 – sin θ) (sec θ + tan θ) = 1
हल:
L.H.S = sec θ(1 – sin θ) (sec θ + tan θ)
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 4

प्रश्न 5.
cosec2θ + sec2θ = cosec2θ sec θ (RBSESolutions.com)
हल:
L.H.S. = cosec2θ + sec2θ
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 5

RBSE Solutions

प्रश्न 6.
\(\sqrt{\frac{1-\sin \theta}{1+\sin \theta}}=\sec \theta-\tan \theta\)
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 6
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 7

प्रश्न 7.
\(\sqrt{\sec ^{2} \theta+\csc ^{2} \theta}\) = tan θ+ cot θ
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 8

प्रश्न 8.
\(\frac{\tan \alpha+\tan \beta}{\cot \alpha+\cot \beta}\) = tan α tan β
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 9
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 9a

प्रश्न 9.
\(\frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}=2 \sec \theta\)
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 10

प्रश्न 10.
\(\frac{\sin ^{4} \theta-\cos ^{4} \theta}{\sin ^{2} \theta-\cos ^{2} \theta}=1\)
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 11

RBSE Solutions

प्रश्न 11.
cot θ – tan θ = \(\frac{1-2 \sin ^{2} \theta}{\sin \theta \cos \theta}\)
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 12

प्रश्न 12.
cose4 θ + sin4 θ = 1 – 2 cos2θ sin2θ (माध्य शिक्षा बोर्ड मॉडल (RBSESolutions.com) पेपर, 2017-18)
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 13

प्रश्न 13.
(sec θ – cos θ) (cot θ + tan θ) = tan θ sec θ
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 14

प्रश्न 14.
\(\frac{1-\tan ^{2} \alpha}{\cot ^{2} \alpha-1}=\tan ^{2} \alpha\)
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 15

प्रश्न 15.
\(\frac{\sin \theta}{1-\cos \theta}=\frac{1+\cos \theta}{\sin \theta}\)
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 16

RBSE Solutions

प्रश्न 16.
sin6θ + cos6θ = 1 – 3 sin2θ cos2θ
हल:
L.H.S = sin6θ + cos6θ
= (sin2θ)3 + (cos2θ)3
= (sin2θ + cos2θ)[(sin2θ)2 + (cos2θ)2 – sin2θ cos2θ]
=(1) [sin4θ + cos4θ – sin2θ cos2θ]
= [(sin2θ)2 + (cos2θ)2 + 2 sin2θ cos2θ – 2 sin2θcos2θ – sin2θ cos2θ]
= [(sin2θ + cos2θ)2 – 3sin2θcos2θ]
= (1)2 – 3 sin2θcos2θ
= 1 – 3 sin2θ cos2θ = R.H.S
∴ L.H.S = R.H.S ( इतिसिद्धम् )

प्रश्न 17.
\(\frac{\tan \theta}{1-\cot \theta}+\frac{\cot \theta}{1-\tan \theta}\) = 1 + tanθ + cotθ (माध्य (RBSESolutions.com) शिक्षा बोर्ड, 2018)
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 17
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 18
= 1 + tan θ + cot θ = R.H.S
∴ L.H.S = R.H.S ( इतिसिद्धम् )

प्रश्न 18.
sin θ (1 + tan θ) + cos θ(1 + cot θ) = cosec θ + sec θ
हल:
L.H.S. = sin θ(1 + tan θ) + cos θ(1 + cot θ)
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 19

प्रश्न 19.
sin2θ cos θ + tan θ sin θ + cos3θ = sec θ
हल:
L.H.S. = sin2θ cos θ + tan θ sin θ + cos23 θ
= sin2θ cos θ + cos3θ + tan θ. sin θ
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 20
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 21

RBSE Solutions

प्रश्न 20.
\(\frac{\tan \theta}{1-\cot \theta}+\frac{\cot \theta}{1-\tan \theta}\) = 1 + sec θ cosec θ
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 22
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 23
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 24

प्रश्न 21.
(sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2A + cot2A
हल:
L.H.S. = (sin A + cosec A)2 + (cos A + sec A)2
= {sin2 A + cosec2 A + 2 sin A × cosec A}
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 25

प्रश्न 22.
sin8θ – cos8θ = (sin2θ – cos2θ) (1 – 2sin2θ cos2θ)
हल:
L.H.S. = sin8θ – cos8θ
= (sin4θ)2 – (cos4θ)2
= (sin4θ – cos4θ) (sin4θ + cos4θ)
= [(sin2θ)2 – (cos2θ)]2 (sin4θ + cos4θ)
= (sin2θ – cos2θ) (sin2θ + cos2θ) (sin4θ + cos4θ)
= (sin2θ – cos2θ) (sin4θ + cos4θ] [:: sin2θ+ cos2θ = 1]
= (sin2θ – cos2θ) [(sin2θ)2 + (cos2θ)2 + 2sin2θcos2θ– 2sin2θ cos2θ]
(2 sin2θ cos2θ उपर्युक्त मे जोड़ने व घटाने पर)
= (sin2θ – cos2θ) [(sin2θ + cos2θ)2 – 2 sin2θ cos2θ]
= (sin2θ – cos2θ) (1 – 2 sin2θ cos2θ)
= R.H.S.
∴ L.H.S. = R.H.S. ( इतिसिद्धम् )

प्रश्न 23.
\(\sqrt{\frac{\sec \theta+1}{\sec \theta-1}}\) = cot θ + cosec θ (माध्य शिक्षा (RBSESolutions.com) बोर्ड मॉडल पेपर, 2017-18)
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 26

प्रश्न 24.
\(\frac{(1+\cot \theta+\tan \theta)(\sin \theta-\cos \theta)}{\sec ^{3} \theta-\csc ^{3} \theta}=\sin ^{2} \theta \cos ^{2} \theta\)
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 27
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 28

RBSE Solutions

प्रश्न 25.
\(\frac{\sin \theta+\cos \theta}{\sin \theta-\cos \theta}+\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta}=\frac{2}{1-2 \cos ^{2} \theta}=\frac{2}{2 \sin ^{2} \theta-1}\)
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 29
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 30

प्रश्न 26.
\(\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}\) = sin A + cos A (माध्य (RBSESolutions.com) शिक्षा बोर्ड, 2018)
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 31

प्रश्न 27.
(cosec A – sin A)(sec A – cos A) = \(\frac{1}{\tan A+\cot A}\)
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 32
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 33

प्रश्न 28.
\(\frac{\cos ^{2} \theta}{1-\tan \theta}+\frac{\sin ^{3} \theta}{\sin \theta-\cos \theta}\) = 1 + sin θ cos θ
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 34
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 35

प्रश्न 29.
यदि sec θ + tan θ = P हो, तो सिद्ध करो कि \(\frac{\mathbf{P}^{2}-1}{\mathbf{P}^{2}+1}=\sin \theta\)
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 36

RBSE Solutions

प्रश्न 30.
यदि \(\frac{\cos A}{\cos B}=m\) तथा \(\frac{\cos A}{\sin B}=n\) हो, (RBSESolutions.com) तो सिद्ध कीजिए (m2 +n2) cos2B = n2 (माध्य. शिक्षा बोर्ड, मॉडल पेपर, 2017-18 )
हल:
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 37
RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 38

We hope the RBSE Solutions for Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Ex 7.1 help you. If you have any query regarding Rajasthan Board RBSE Class 10 Maths Chapter 7 त्रिकोणमितीय सर्वसमिकाएँ Exercise 7.1, drop a comment below and we will get back to you at the earliest.

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