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

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Question 31 (a) During a math class, Ms. Isha wrote the expression given below on the board and asked the students to simplify it. (cos⁡𝜃)/(1−sin⁡𝜃)+(1−sin⁡𝜃)/(cos⁡𝜃) Jyoti solved it in her note book as follows: Identify the step in which Jyoti has made error(s), if any. Rectify the same and find the correct answer. cos 8 1-sin 8 1-—sin 0 cos 8 _ cos*8+(1—sin@)? ~ (1=sin @)xcos®- _ _cos*8+cos*6 7 (1-sin 8)xcos@ 2cos70 a (1-sin 8)xcos®@ _ _2cos8 ~ (1-sin @) .. (Step 1) .. (Step 2) .. (Step 3) .. (step 4) Question 31 (a) During a math class, Ms. Isha wrote the expression given below on the board and asked the students to simplify it. (cos⁡𝜃)/(1−sin⁡𝜃)+(1−sin⁡𝜃)/(cos⁡𝜃) Jyoti solved it in her note book as follows: [, , (cos⁡𝜃)/(1−sin⁡𝜃)+(1−sin⁡𝜃)/(cos⁡𝜃); =, (cos^(2)⁡𝜃+(1−sin⁡𝜃)^(2))/((1−sin⁡𝜃)×cos⁡𝜃), (step1); =, (cos^(2)⁡𝜃+cos^(2)⁡𝜃)/((1−sin⁡𝜃)×cos⁡𝜃), (step2); =, (2cos^(2)⁡𝜃)/((1−sin⁡𝜃)×cos⁡𝜃), (step2); =, (2cos⁡𝜃)/((1−sin⁡𝜃)), (step2)] Identify the step in which Jyoti has made error(s), if any. Rectify the same and find the correct answer. The error is in step 2 She put (1−𝑠𝑖𝑛⁡𝜃)^(2) = 𝑐𝑜𝑠^(2)𝜃 But, 1−𝑠𝑖𝑛^(2)𝜃=𝑐𝑜𝑠^(2)𝜃 Solving again (cos⁡𝜃)/(1−sin⁡𝜃)+(1 − sin⁡𝜃)/(cos⁡𝜃) = (cos^(2)𝜃 + (𝟏 − 𝐬𝐢𝐧⁡𝜽)^(𝟐))/((1 − sin𝜃) × cos⁡𝜃) = (cos^(2)𝜃 + 1^(2) + sin^(2) 𝜃 −2 sin 𝜃 )/((1 − sin𝜃) × cos⁡𝜃) = (𝒄𝒐𝒔^(𝟐)𝜽 + 𝒔𝒊𝒏^(𝟐) 𝜽 + 1 − 2 sin 𝜃 )/((1 − sin𝜃) × cos⁡𝜃) Putting cos^(2)𝜃 + sin^(2) 𝜃=1 = (𝟏 + 1 − 2 sin 𝜃 )/((1 − sin𝜃) × cos⁡𝜃) = (2 − 2 𝑠𝑖𝑛 𝜃 )/((1 − 𝑠𝑖𝑛𝜃) × 𝑐𝑜𝑠⁡𝜃) = (2(1 − 𝑠𝑖𝑛𝜃))/((1 − 𝑠𝑖𝑛𝜃) × 𝑐𝑜𝑠⁡𝜃) = (2)/(𝑐𝑜𝑠⁡𝜃) = 2 sec 𝜃 = (2 − 2 𝑠𝑖𝑛 𝜃 )/((1 − 𝑠𝑖𝑛𝜃) × 𝑐𝑜𝑠⁡𝜃) = (2(1 − 𝑠𝑖𝑛𝜃))/((1 − 𝑠𝑖𝑛𝜃) × 𝑐𝑜𝑠⁡𝜃) = (2)/(𝑐𝑜𝑠⁡𝜃) = 2 sec 𝜃