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We have certain trigonometric identities.   Like, \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\]  and  \[1+{{\tan }^{2}}\theta ={{\sec }^{2}}\theta \] etc.   Such identities are identities in the sense that they hold for all value of the angles which satisfy the given condition among them and they are called conditional identities.   (1) If \[A+B+C={{180}^{o}}\], then   (i) \[\sin 2A+\sin 2B+\sin 2C=4\sin A\sin B\sin C\]   (ii) \[\sin 2A+\sin 2B-\sin 2C=4\cos A\cos B\sin C\]   (iii) \[\sin (B+C-A)+\sin (C+A-B)+\sin (A+B-C)\]\[=4\sin A\sin B\sin C\]   (iv) \[\cos 2A+\cos 2B+\cos 2C=-1-4\cos A\cos B\cos C\]   (v) \[\cos 2A+\cos 2B-\cos 2C=1-4\sin A\sin B\cos C\]   (2) If \[A+B+C={{180}^{o}}\], then   (i) \[\sin A+\sin B+\sin C=4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}\]   (ii) \[\sin A+\sin B-\sin C=4\sin \frac{A}{2}\sin \frac{B}{2}\cos \frac{C}{2}\]   (iii) \[\cos A+\cos B+\cos C=1+4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\]   (iv) \[\cos A+\cos B-\cos C=-1+4\cos \frac{A}{2}\cos \frac{B}{2}\sin \frac{C}{2}\]   (v) \[\frac{\cos A}{\sin B\sin C}+\frac{\cos B}{\sin C\sin A}+\frac{\cos C}{\sin A\sin B}=2\]   (3) If \[A+B+C=\pi \], then   (i) \[{{\sin }^{2}}A+{{\sin }^{2}}B-{{\sin }^{2}}C=2\sin A\sin B\cos C\]   (ii) \[{{\cos }^{2}}A+{{\cos }^{2}}B+{{\cos }^{2}}C=1-2\cos A\cos B\cos C\]   (iii) \[{{\sin }^{2}}A+{{\sin }^{2}}B+{{\sin }^{2}}C=1-2\sin A\sin B\cos C\]   (4) If \[A+B+C=\pi ,\] then   (i) \[{{\sin }^{2}}\frac{A}{2}+{{\sin }^{2}}\frac{B}{2}+{{\sin }^{2}}\frac{C}{2}=1-2\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\]   (ii) \[{{\cos }^{2}}\frac{A}{2}+{{\cos }^{2}}\frac{B}{2}+{{\cos }^{2}}\frac{C}{2}=2+2\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\]   (iii) \[{{\sin }^{2}}\frac{A}{2}+{{\sin }^{2}}\frac{B}{2}-{{\sin }^{2}}\frac{C}{2}=1-2\cos \frac{A}{2}\cos \frac{B}{2}\sin \frac{C}{2}\]   (iv) \[{{\cos }^{2}}\frac{A}{2}+{{\cos }^{2}}\frac{B}{2}-{{\cos }^{2}}\frac{C}{2}=2\cos \frac{A}{2}\cos \frac{B}{2}\sin \frac{C}{2}\]   (5) If \[x+y+z=\frac{\pi }{2}\], then   (i) \[{{\sin }^{2}}x+{{\sin }^{2}}y+{{\sin }^{2}}z=1-2\sin x\sin y\sin z\]   (ii) \[{{\cos }^{2}}x+{{\cos }^{2}}y+{{\cos }^{2}}z=2+2\sin x\sin y\sin z\]   (iii) \[\sin 2x+\sin 2y+\sin 2z=4\cos x\cos y\cos z\]   (6) If \[A+B+C=\pi \], then   (i) \[\tan A+\tan B+\tan C=\tan A\tan B\tan C\]   (ii) \[\cot B\cot C+\cot C\cot A+\cot A\cot B=1\]   (iii) \[\tan \frac{B}{2}\tan \frac{C}{2}+\tan \frac{C}{2}\tan \frac{A}{2}+\tan \frac{A}{2}\tan \frac{B}{2}=1\]   (iv) \[\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}=\cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}\]

A sequence of numbers \[<{{t}_{n}}>\] is said to be in arithmetic progression (A.P.) when the difference \[{{t}_{n}}-{{t}_{n-1}}\] is a constant for all n Î N. This constant is called the common difference of the A.P. and is usually denoted by the letter d.     If \['a'\] is the first term and \['d'\] the common difference, then an A.P. can be represented as \[a,\,a+d,a+2d,\,a+3d,........\]     Example : 2, 7, 12, 17, 22, …… is an A.P. whose first term is 2 and common difference 5.     Algorithm to determine whether a sequence is an A.P. or not.     Step I: Obtain \[{{a}_{n}}\] (the \[{{n}^{th}}\] term of the sequence).     Step II: Replace \[n\] by \[n-1\] in \[{{a}_{n}}\] to get \[{{a}_{n-1}}\].     Step III: Calculate \[{{a}_{n}}-{{a}_{n-1}}\].     If \[{{a}_{n}}-{{a}_{n-1}}\] is independent from \[n,\] the given sequence is an A.P. otherwise it is not an A.P.     \[\therefore \] \[{{t}_{n}}=An+B\] represents the \[{{n}^{th}}\] term of an A.P. with common difference A.

(1) Let \['a'\] be the first term and \['d'\] be the common difference of an A.P. Then its \[{{n}^{th}}\] term is \[a+(n-1)d\]i.e., \[{{T}_{n}}=a+(n-1)d\].   (2) \[{{p}^{th}}\] term of an A.P. from the end : Let \['a'\] be the first term and \['d'\] be the common difference of an A.P. having \[n\] terms. Then  \[{{p}^{th}}\] term from the end is \[{{(n-p+1)}^{th}}\] term from the beginning   i.e., \[{{p}^{th}}\text{ term from the end }=\text{ }{{T}_{(n-p+1)}}=a+(n-p)d\].  

• If last term of an A.P. is l then \[{{p}^{th}}\]term from end\[=l-(p-1)d\]

  When the sum is given, the following way is adopted in selecting certain number of terms :    

Number of terms	Terms to be taken
3	\[ad,a,a+d\]
4	\[a\text{ }3d,ad,a+d,a+\text{ }3d\]
5	\[a\text{ }2d,ad,a,a+d,a+\text{ }2d\]

    In general, we take \[ard,a(r1)d,\,\,......,\,\,ad,a,a+d,\text{ }\ldots \ldots ,a+\text{ }(r1)d,\,\,a+rd,\] in case we have to take \[(2r+1)\] terms (i.e. odd number of terms) in an A.P.     And,\[a-(2r-1)d,\,a-(2r-3)d,\,.......,\] \[\,a-d,\,a+d,\,.......,\,\,a+(2r-1)d\]  in case we have to take \[2r\] terms in an A.P.     When the sum is not given, then the following way is adopted in selection of terms.      

Number of terms	Terms to be taken
3	\[a,\,a+d,\,a+2d\]
4	\[a,\,a+d,\,a+2d,\,a+3d\]
5	\[a,\,a+d,\,a+2d,\,a+3d,\,a+4d\]

   

  The sum of n terms of the series   \[a+(a+d)+(a+2d)+.......+\{a+(n-1)\,d\}\] is given by   \[{{S}_{n}}=\frac{n}{2}[2a+(n-1)\,d]\]   Also, \[{{S}_{n}}=\frac{n}{2}(a+l)\], where \[l=\] last term \[=a+(n-1)d\].  

  If \[a,A,b\] are in A.P., then A is called A.M. between \[a\] and \[b\].   (1) If \[a,\,{{A}_{1}},\,{{A}_{2}},\,{{A}_{3}},.....,\,{{A}_{n}},\,b\] are in A.P., then \[{{A}_{1}},\,{{A}_{2}},\,{{A}_{3}},\,......,\,{{A}_{n}}\] are called \[n\] A.M.?s between \[a\] and \[b\].   (2) Insertion of arithmetic means   (i) Single A.M. between \[a\] and \[b\] : If \[a\] and \[b\] are two real numbers then single A.M. between \[a\] and \[b\]\[=\frac{a+b}{2}\]   (ii) n A.M.?s between a and b : If \[{{A}_{1}},\,{{A}_{2}},\,{{A}_{3}},\,.......,\,{{A}_{n}}\] are n A.M.?s between \[a\] and \[b\], then   \[{{A}_{1}}=a+d=a+\frac{b-a}{n+1}\], \[{{A}_{2}}=a+2d=a+2\frac{b-a}{n+1}\],   \[{{A}_{3}}=a+3d=a+3\frac{b-a}{n+1}\], ??., \[{{A}_{n}}=a+nd=a+n\frac{b-a}{n+1}\].

  (1) If \[{{a}_{1}},\,{{a}_{2}},\,{{a}_{3}}.....\] are in A.P. whose common difference is \[d,\] then for fixed non-zero number \[k\in R\].   (i) \[{{a}_{1}}\pm k,\,{{a}_{2}}\pm k,\,{{a}_{3}}\pm k,.....\] will be in A.P., whose common difference will be \[d\].   (ii) \[k{{a}_{1}},\,k{{a}_{2}},\,k{{a}_{3}}....\] will be in A.P. with common difference \[=kd\].   (iii) \[\frac{{{a}_{1}}}{k},\,\frac{{{a}_{2}}}{k},\,\frac{{{a}_{3}}}{k}......\] will be in A.P. with common difference \[=d/k\].   (2) The sum of terms of an A.P. equidistant from the beginning and the end is constant and is equal to sum of first and last term. i.e. \[{{a}_{1}}+{{a}_{n}}={{a}_{2}}+{{a}_{n-1}}={{a}_{3}}+{{a}_{n-2}}=....\]   (3) If number of terms of any A.P. is odd, then sum of the terms is equal to product of middle term and number of terms.   (4) If number of terms of any A.P. is even then A.M. of middle two terms is A.M. of first and last term.   (5) If the number of terms of an A.P. is odd then its middle term is A.M. of first and last term.   (6) If \[{{a}_{1}},\,{{a}_{2}},\,......{{a}_{n}}\] and \[{{b}_{1}},\,{{b}_{2}},\,......{{b}_{n}}\] are the two A.P.'s. Then \[{{a}_{1}}\pm {{b}_{1}},\,{{a}_{2}}\pm {{b}_{2}},\,......{{a}_{n}}\pm {{b}_{n}}\] are also A.P.'s with common difference \[{{d}_{1}}\ne {{d}_{2}}\], where \[{{d}_{1}}\] and \[{{d}_{2}}\] are the common difference of the given A.P.'s.   (7) Three numbers \[a,\,\,b,\,\,\,c\] are in A.P. iff \[2b=a+c\].   (8) If \[{{T}_{n}},\,{{T}_{n+1}}\] and \[{{T}_{n+2}}\] are three consecutive terms of an A.P., then \[2{{T}_{n+1}}={{T}_{n}}+{{T}_{n+2}}\].   (9) If the terms of an A.P. are chosen at regular intervals, then they form an A.P.  

  A progression is called a G.P. if the ratio of its each term to its previous term is always constant. This constant ratio is called its common ratio and it is generally denoted by \[r\].   Example: The sequence 4, 12, 36, 108, ?.. is a G.P., because \[\frac{12}{4}=\frac{36}{12}=\frac{108}{36}=.....=3\], which is constant.   Clearly, this sequence is a G.P. with first term 4 and common ratio 3.   The sequence \[\frac{1}{3},\,-\frac{1}{2},\,\frac{3}{4},\,-\frac{9}{8},\,....\] is a G.P. with first term \[\frac{1}{3}\] and common ratio \[{\left( -\frac{1}{2} \right)}/{\left( \frac{1}{3} \right)=-\frac{3}{2}}\;\].  

(1) We know that, \[a,\,ar,\,a{{r}^{2}},\,a{{r}^{3}},\,.....a{{r}^{n-1}}\] is a sequence of G.P.   Here, the first term is ‘a’ and the common ratio is \['r'\].   The general term or \[{{n}^{th}}\] term of a G.P. is \[{{T}_{n}}=a{{r}^{n-1}}\].   It should be noted that, \[r=\frac{{{T}_{2}}}{{{T}_{1}}}=\frac{{{T}_{3}}}{{{T}_{2}}}=......\].   (2) \[{{p}^{th}}\] term from the end of a finite G.P. : If G.P. consists of \['n'\] terms, \[{{p}^{th}}\] term from the end \[={{(n-p+1)}^{th}}\] term from the beginning \[=a{{r}^{n-p}}\].   Also, the \[{{p}^{th}}\] term from the end of a G.P. with last term \[l\]and common ratio \[r\] is \[l\,{{\left( \frac{1}{r} \right)}^{n-1}}\].

  (1) When the product is given, the following way is adopted in selecting certain number of terms :    

Number of terms	Terms to be taken
3	\[\frac{a}{r},\,a,\,ar\]
4	\[\frac{a}{{{r}^{3}}},\,\frac{a}{r},\,ar,\,a{{r}^{3}}\]
5	\[\frac{a}{{{r}^{2}}},\,\,\frac{a}{r},\,\,a,\,\,ar,\,\,a{{r}^{2}}\]

    (2) When the product is not given, then the following way is adopted in selection of terms        

Number of terms	Terms to be taken
3	\[a,\,\,ar,\,\,a{{r}^{2}}\]
4	\[a,\,\,ar,\,\,a{{r}^{2}},\,a{{r}^{3}}\]
5	\[a,\,\,ar,\,\,a{{r}^{2}},\,a{{r}^{3}},\,a{{r}^{4}}\]