Question 31 (b) - CBSE Class 10 Sample Paper for 2027 Boards - Maths Standard - Solutions of Sample Papers for Class 10 Boards

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Question 31 (b) If (1)/(sin𝑥 − cos⁡𝑥)=(cosec 𝑥)/(sqrt(2)), then prove that ((1)/(sin𝑥 + cos⁡𝑥))^(2)=(sec^(2)⁡𝑥)/(2) Given (1)/(sin𝑥 − cos⁡𝑥)=(cosec 𝑥)/(sqrt(2)) (sqrt(𝟐))/(𝒄𝒐𝒔𝒆𝒄 𝒙)=𝒔𝒊𝒏 𝒙−𝒄𝒐𝒔 𝒙 (sqrt(2))/((1)/(sin𝑥))=𝑠𝑖𝑛 𝑥−𝑐𝑜𝑠 𝑥 sqrt(2) ×(sin𝑥)/(1)=𝑠𝑖𝑛 𝑥−𝑐𝑜𝑠 𝑥 sqrt(2) ×(sin𝑥)/(1)=𝑠𝑖𝑛 𝑥−𝑐𝑜𝑠 𝑥 𝒔𝒊𝒏 𝒙−𝒄𝒐𝒔 𝒙=sqrt(𝟐)𝐬𝐢𝐧𝒙 Squaring both sides (𝑠𝑖𝑛 𝑥−𝑐𝑜𝑠 𝑥)^(2)=(sqrt(2)sin𝑥)^(2) 𝐬𝐢𝐧^(𝟐)𝒙+𝐜𝐨𝐬^(𝟐)𝒙−𝟐𝐬𝐢𝐧𝒙𝐜𝐨𝐬𝒙=𝟐𝐬𝐢𝐧^(𝟐)𝒙 Putting sin^(2)𝑥+cos^(2)𝑥=1 𝟏−𝟐𝒔𝒊𝒏𝒙𝒄𝒐𝒔𝒙=𝟐𝒔𝒊𝒏^(𝟐)𝒙 1−2sin^(2)𝑥=2sin𝑥cos𝑥 𝟐𝒔𝒊𝒏𝒙𝒄𝒐𝒔𝒙=𝟏−𝟐𝒔𝒊𝒏^(𝟐)𝒙 Now, we need to prove ((𝟏)/(𝒔𝒊𝒏𝒙 + 𝐜𝐨𝐬⁡𝒙))^(𝟐)=(𝒔𝒆𝒄^(𝟐)⁡𝒙)/(𝟐) Solving LHS ((1)/(𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠⁡𝑥))^(2) =(1)/(sin^(2)𝑥+cos^(2)𝑥+2sin𝑥cos𝑥) Putting sin^(2)𝑥+cos^(2)𝑥=1 =(1)/(sin^(2)𝑥+cos^(2)𝑥+2sin𝑥cos𝑥) (𝑠𝑖𝑛 𝑥−𝑐𝑜𝑠 𝑥)^(2)=(sqrt(2)sin𝑥)^(2) 𝐬𝐢𝐧^(𝟐)𝒙+𝐜𝐨𝐬^(𝟐)𝒙−𝟐𝐬𝐢𝐧𝒙𝐜𝐨𝐬𝒙=𝟐𝐬𝐢𝐧^(𝟐)𝒙 Putting sin^(2)𝑥+cos^(2)𝑥=1 𝟏−𝟐𝒔𝒊𝒏𝒙𝒄𝒐𝒔𝒙=𝟐𝒔𝒊𝒏^(𝟐)𝒙 1−2sin^(2)𝑥=2sin𝑥cos𝑥 𝟐𝒔𝒊𝒏𝒙𝒄𝒐𝒔𝒙=𝟏−𝟐𝒔𝒊𝒏^(𝟐)𝒙 Now, we need to prove ((𝟏)/(𝒔𝒊𝒏𝒙 + 𝐜𝐨𝐬⁡𝒙))^(𝟐)=(𝒔𝒆𝒄^(𝟐)⁡𝒙)/(𝟐) Solving LHS ((1)/(𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠⁡𝑥))^(2) =(1)/(sin^(2)𝑥 + cos^(2)𝑥 + 2sin𝑥cos𝑥) Putting sin^(2)𝑥+cos^(2)𝑥=1 =(𝟏)/(𝟏 + 𝟐𝒔𝒊𝒏𝒙𝒄𝒐𝒔𝒙) Putting 2𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥=1−2𝑠𝑖𝑛^(2)𝑥 =(𝟏)/(𝟏 +(𝟏 − 𝟐 𝒔𝒊𝒏^(𝟐)𝒙)) =(1)/(2 − 2 𝑠𝑖𝑛^(2)𝑥) =(𝟏)/(𝟐(𝟏 − 𝒔𝒊𝒏^(𝟐)𝒙)) Putting 1−sin^(2)𝑥=cos^(2)𝑥 =(1)/(2𝑐𝑜𝑠^(2) 𝑥) =(𝐬𝐞𝐜^(𝟐)𝒙)/(𝟐) = RHS Thus, LHS = RHS Hence proved