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أثبت أن المعادلة الآتية تمثل متطابقة :

cotθcotθ cos 2 θ=cotθ sin 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaWYbqqaaaa aaOpGqSvxza8qaciGGJbGaai4BaiaacshacqaH4oqCcqGHsislciGG JbGaai4BaiaacshacqaH4oqCciGGJbGaai4BaiaacohadaahaaWcbe qaaiaaikdaaaGccqaH4oqCcqGH9aqppaGaci4yaiaac+gacaGG0bGa eqiUdeNaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeq iUdehaaa@5360@    
cotθcotθ cos 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaOpGqSv xza8qabiqaaWYbciGGJbGaai4BaiaacshacqaH4oqCcqGHsislciGG JbGaai4BaiaacshacqaH4oqCciGGJbGaai4BaiaacohadaahaaWcbe qaaiaaikdaaaGccqaH4oqCaaa@4840@                      
 
cotθ(1 cos 2 θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaOpGqSv xza8qabiqaaWYbciGGJbGaai4BaiaacshacqaH4oqCcaGGOaGaaGym aiabgkHiTiGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaaki abeI7aXjaacMcaaaa@45CA@ =    
 

cotθcotθ cos 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaWYbciGGJb Gaai4BaiaacshacqaH4oqCcqGHsislciGGJbGaai4BaiaacshacqaH 4oqCciGGJbGaai4BaiaacohadaahaaWcbeqaaiaaikdaaaGccqaH4o qCaaa@45A9@

=      

1cosθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaWYbcaaIXa GaeyOeI0Iaci4yaiaac+gacaGGZbGaeqiUdehaaa@3C5D@

=    
 
sinθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaWYbqqaaaa aaOpGqSvxza8qaciGGZbGaaiyAaiaac6gacqaH4oqCaaa@3D51@

=

     
         
cotθ sin 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGG0bGaeqiUdeNaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaI YaaaaOGaeqiUdehaaa@4001@      
 
cosθ sinθ sin 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaG1cdaWcaa qaaiGacogacaGGVbGaai4CaiabeI7aXbqaaiGacohacaGGPbGaaiOB aiabeI7aXbaacqGHIaYTciGGZbGaaiyAaiaac6gadaahaaWcbeqaai aaikdaaaGccqaH4oqCaaa@4689@ =    
 

sinθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaWYbqqaaaa aaOpGqSvxza8qaciGGZbGaaiyAaiaac6gacqaH4oqCaaa@3D51@

=      

الطرفان متساويان

       
cotθcotθ cos 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaOpGqSv xza8qabiqaaWYbciGGJbGaai4BaiaacshacqaH4oqCcqGHsislciGG JbGaai4BaiaacshacqaH4oqCciGGJbGaai4BaiaacohadaahaaWcbe qaaiaaikdaaaGccqaH4oqCaaa@4840@                      
 
cotθ(1 cos 2 θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaOpGqSv xza8qabiqaaWYbciGGJbGaai4BaiaacshacqaH4oqCcaGGOaGaaGym aiabgkHiTiGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaaki abeI7aXjaacMcaaaa@45CA@ =    
 

cotθsin θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaWYbqqaaaa aaOpGqSvxza8qaciGGJbGaai4BaiaacshacqaH4oqCcqGHIaYTciGG ZbGaaiyAaiaac6gadaahbaWcbeqaaiaaikdaaaGccqaH4oqCaaa@4454@

=      

cosθ sinθ sin θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaWYbqqaaaa aaOpGqSvxza8qadaWcaaqaaiGacogacaGGVbGaai4CaiabeI7aXbqa aiGacohacaGGPbGaaiOBaiabeI7aXbaacqGHIaYTciGGZbGaaiyAai aac6gadaahbaWcbeqaaiaaikdaaaGccqaH4oqCaaa@48F1@

=    
 
cosθsinθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaOpGqSv xza8qabiqaaWYbciGGJbGaai4BaiaacohacqaH4oqCciGGZbGaaiyA aiaac6gacqaH4oqCaaa@41DA@

=

     
         
cotθ sin 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGG0bGaeqiUdeNaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaI YaaaaOGaeqiUdehaaa@4001@      
 
cosθ sinθ sin 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaG1cdaWcaa qaaiGacogacaGGVbGaai4CaiabeI7aXbqaaiGacohacaGGPbGaaiOB aiabeI7aXbaacqGHIaYTciGGZbGaaiyAaiaac6gadaahaaWcbeqaai aaikdaaaGccqaH4oqCaaa@4689@ =    
 

cosθsinθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqa6daaaaaGuLr gapeqaaiGacogacaGGVbGaai4CaiabeI7aXjGacohacaGGPbGaaiOB aiabeI7aXbaa@4137@

=      

الطرفان متساويان

       
cotθcotθ cos 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaOpGqSv xza8qabiqaaWYbciGGJbGaai4BaiaacshacqaH4oqCcqGHsislciGG JbGaai4BaiaacshacqaH4oqCciGGJbGaai4BaiaacohadaahaaWcbe qaaiaaikdaaaGccqaH4oqCaaa@4840@                      
 
cotθ(1 cos 2 θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaOpGqSv xza8qabiqaaWYbciGGJbGaai4BaiaacshacqaH4oqCcaGGOaGaaGym aiabgkHiTiGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaaki abeI7aXjaacMcaaaa@45CA@ =    
 

cotθsinθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4HqaqFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaG1cqqaaaa aaOpGqSvxza8qaciGGJbGaai4BaiaacshacqaH4oqCcqGHIaYTciGG ZbGaaiyAaiaac6gacqaH4oqCaaa@4341@

=      

cosθ sinθ sinθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaWYbqqaaaa aaOpGqSvxza8qadaWcaaqaaiGacogacaGGVbGaai4CaiabeI7aXbqa aiGacohacaGGPbGaaiOBaiabeI7aXbaacqGHIaYTciGGZbGaaiyAai aac6gacqaH4oqCaaa@47FD@

=    
 
cosθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaWYbqqaaaa aaOpGqSvxza8qaciGGJbGaai4BaiaacohacqaH4oqCaaa@3D4C@

=

     
         
cotθ sin 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGG0bGaeqiUdeNaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaI YaaaaOGaeqiUdehaaa@4001@      
 
cosθ sin 2 θ sin 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaG1cdaWcaa qaaiGacogacaGGVbGaai4CaiabeI7aXbqaaiGacohacaGGPbGaaiOB amaaCaaaleqabaGaaGOmaaaakiabeI7aXbaacqGHIaYTciGGZbGaai yAaiaac6gadaahaaWcbeqaaiaaikdaaaGccqaH4oqCaaa@477C@ =    
 

cosθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaWYbqqaaaa aaOpGqSvxza8qaciGGJbGaai4BaiaacohacqaH4oqCaaa@3D4C@

=      

الطرفان متساويان

       
            
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